Internal problem ID [13066]
Internal file name [OUTPUT/11722_Sunday_December_03_2023_07_15_19_PM_78487314/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 3. Linear Systems. Exercises section 3.1. page 258
Problem number: 7.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs"
Solve \begin {align*} p^{\prime }\left (t \right )&=3 p \left (t \right )-2 q \left (t \right )-7 r \left (t \right )\\ q^{\prime }\left (t \right )&=-2 p \left (t \right )+6 r \left (t \right )\\ r^{\prime }\left (t \right )&=\frac {73 q \left (t \right )}{100}+2 r \left (t \right ) \end {align*}
In this method, we will assume we have found the matrix exponential \(e^{A t}\) allready. There are different methods to determine this but will not be shown here. This is a system of linear ODE’s given as Warning. Unable to find the matrix exponential.
This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) \end {align*}
Or \begin {align*} \left [\begin {array}{c} p^{\prime }\left (t \right ) \\ q^{\prime }\left (t \right ) \\ r^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} 3 & -2 & -7 \\ -2 & 0 & 6 \\ 0 & \frac {73}{100} & 2 \end {array}\right ]\, \left [\begin {array}{c} p \left (t \right ) \\ q \left (t \right ) \\ r \left (t \right ) \end {array}\right ] \end {align*}
The first step is find the homogeneous solution. We start by finding the eigenvalues of \(A\). This is done by solving the following equation for the eigenvalues \(\lambda \) \begin {align*} \operatorname {det} \left ( A- \lambda I \right ) &= 0 \end {align*}
Expanding gives \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} 3 & -2 & -7 \\ -2 & 0 & 6 \\ 0 & \frac {73}{100} & 2 \end {array}\right ]-\lambda \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) &= 0 \end {align*}
Therefore \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} 3-\lambda & -2 & -7 \\ -2 & -\lambda & 6 \\ 0 & \frac {73}{100} & 2-\lambda \end {array}\right ]\right ) &= 0 \end {align*}
Which gives the characteristic equation \begin {align*} \lambda ^{3}-5 \lambda ^{2}-\frac {119}{50} \lambda +\frac {273}{25}&=0 \end {align*}
The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= \frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\\ \lambda _2 &= -\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
This table summarises the above result
| eigenvalue | algebraic multiplicity | type of eigenvalue |
| \(-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | complex eigenvalue |
| \(\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\) | \(1\) | complex eigenvalue |
| \(-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | complex eigenvalue |
Now the eigenvector for each eigenvalue are found.
Considering the eigenvalue \(\lambda _{1} = \frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} 3 & -2 & -7 \\ -2 & 0 & 6 \\ 0 & \frac {73}{100} & 2 \end {array}\right ] - \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}} & -2 & -7 \\ -2 & \frac {-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-50 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}} & 6 \\ 0 & \frac {73}{100} & \frac {-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {4}{3}-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&-2&-7&0\\ -2&-\frac {5}{3}-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&6&0\\ 0&{\frac {73}{100}}&\frac {1}{3}-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}+\frac {2 R_{1}}{\frac {4}{3}-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&-2&-7&0\\ 0&\frac {-i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-10 i \sqrt {895302429}-138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-1773516}{5 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )}&\frac {6 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+180 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+19284}{\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214}&0\\ 0&{\frac {73}{100}}&\frac {-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {73 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right ) R_{2}}{20 \left (-i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-10 i \sqrt {895302429}-138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-1773516\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&-2&-7&0\\ 0&\frac {-i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-10 i \sqrt {895302429}-138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-1773516}{5 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )}&\frac {6 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+180 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+19284}{\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}} & -2 & -7 \\ 0 & \frac {-i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-10 i \sqrt {895302429}-138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-1773516}{5 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} & \frac {6 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+180 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+19284}{\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = -\frac {30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} t \left (7 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+430 i \sqrt {895302429}+2766 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )}, v_{2} = \frac {60 t \left (3 i \sqrt {895302429}+15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+1607 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+15565\right )}{i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516}\right \}\)
Hence the solution is \[ \left [\begin {array}{c} -\frac {30 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}} t \left (7 \,\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+430 \,\operatorname {I} \sqrt {895302429}+2766 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+10 \,\operatorname {I} \sqrt {895302429}+138 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} \\ \frac {60 t \left (3 \,\operatorname {I} \sqrt {895302429}+15 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+1607 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+15565\right )}{\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+10 \,\operatorname {I} \sqrt {895302429}+138 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+1773516} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} t \left (7 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+430 i \sqrt {895302429}+2766 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} \\ \frac {60 t \left (3 i \sqrt {895302429}+15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+1607 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+15565\right )}{i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} -\frac {30 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}} t \left (7 \,\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+430 \,\operatorname {I} \sqrt {895302429}+2766 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+10 \,\operatorname {I} \sqrt {895302429}+138 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} \\ \frac {60 t \left (3 \,\operatorname {I} \sqrt {895302429}+15 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+1607 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+15565\right )}{\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+10 \,\operatorname {I} \sqrt {895302429}+138 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+1773516} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \left (7 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+430 i \sqrt {895302429}+2766 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} \\ \frac {180 i \sqrt {895302429}+900 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+96420 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+933900}{i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} -\frac {30 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}} t \left (7 \,\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+430 \,\operatorname {I} \sqrt {895302429}+2766 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+10 \,\operatorname {I} \sqrt {895302429}+138 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} \\ \frac {60 t \left (3 \,\operatorname {I} \sqrt {895302429}+15 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+1607 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+15565\right )}{\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+10 \,\operatorname {I} \sqrt {895302429}+138 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+1773516} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \left (7 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+430 i \sqrt {895302429}+2766 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} \\ \frac {180 i \sqrt {895302429}+900 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+96420 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+933900}{i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} -\frac {30 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}} t \left (7 \,\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+430 \,\operatorname {I} \sqrt {895302429}+2766 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+10 \,\operatorname {I} \sqrt {895302429}+138 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} \\ \frac {60 t \left (3 \,\operatorname {I} \sqrt {895302429}+15 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+1607 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+15565\right )}{\operatorname {I} \sqrt {895302429}\, \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+10 \,\operatorname {I} \sqrt {895302429}+138 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 \,\operatorname {I} \sqrt {895302429}\right )^{\frac {1}{3}}+1773516} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \left (7 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+430 i \sqrt {895302429}+2766 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+266655 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+14282412\right )}{\left (i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516\right ) \left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-40 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right )} \\ \frac {180 i \sqrt {895302429}+900 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+96420 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+933900}{i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10 i \sqrt {895302429}+138 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+10545 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+1773516} \\ 1 \end {array}\right ] \] Considering the eigenvalue \(\lambda _{2} = -\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} 3 & -2 & -7 \\ -2 & 0 & 6 \\ 0 & \frac {73}{100} & 2 \end {array}\right ] - \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {4}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}-\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30} & -2 & -7 \\ -2 & -\frac {5}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}-\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30} & 6 \\ 0 & \frac {73}{100} & \frac {1}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}-\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {4}{3}+\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}&-2&-7&0\\ -2&-\frac {5}{3}+\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}&6&0\\ 0&{\frac {73}{100}}&\frac {1}{3}+\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}+\frac {2 R_{1}}{\frac {4}{3}+\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {4}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}-\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}&-2&-7&0\\ 0&\frac {\left (-i \sqrt {895302429}+10545 i \sqrt {3}-3 \sqrt {298434143}-10545\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-1773516 i \sqrt {3}-10 i \sqrt {895302429}+30 \sqrt {298434143}+276 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-1773516}{5 \left (3214+80 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&\frac {19284-360 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+6 i \left (\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214\right ) \sqrt {3}+6 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}}{3214+80 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+i \left (\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214\right ) \sqrt {3}+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}}&0\\ 0&{\frac {73}{100}}&\frac {1}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}-\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {73 \left (3214+80 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} R_{2}}{20 \left (\left (-i \sqrt {895302429}+10545 i \sqrt {3}-3 \sqrt {298434143}-10545\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-1773516 i \sqrt {3}-10 i \sqrt {895302429}+30 \sqrt {298434143}+276 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-1773516\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {4}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}-\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}&-2&-7&0\\ 0&\frac {\left (-i \sqrt {895302429}+10545 i \sqrt {3}-3 \sqrt {298434143}-10545\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-1773516 i \sqrt {3}-10 i \sqrt {895302429}+30 \sqrt {298434143}+276 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-1773516}{5 \left (3214+80 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&\frac {19284-360 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+6 i \left (\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214\right ) \sqrt {3}+6 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}}{3214+80 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+i \left (\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214\right ) \sqrt {3}+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}}&0\\ 0&0&\frac {-37257380160+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \left (1042436862+1821 \left (i \left (\frac {\sqrt {298434143}}{3}-\frac {1819615}{607}\right ) \sqrt {3}-\sqrt {298434143}-\frac {1819615}{607}\right ) \sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )+1821 \left (\frac {5458845 i}{607}+\left (\frac {1819615}{607}+\sqrt {298434143}\right ) \sqrt {3}-i \sqrt {298434143}\right ) \sqrt {3214}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )+2 i \left (-250 \sqrt {298434143}+521218431\right ) \sqrt {3}+1500 \sqrt {298434143}\right )+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \left (2881485+15 \sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right ) \left (i \left (-\frac {\sqrt {298434143}}{3}-\frac {99736}{5}\right ) \sqrt {3}-\sqrt {298434143}+\frac {99736}{5}\right )+15 \left (\frac {299208 i}{5}+\left (-\frac {99736}{5}+\sqrt {298434143}\right ) \sqrt {3}+i \sqrt {298434143}\right ) \sqrt {3214}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )+i \left (607 \sqrt {298434143}-2881485\right ) \sqrt {3}+1821 \sqrt {298434143}\right )+161564700 \left (-1-\frac {3 i \sqrt {3}\, \sqrt {298434143}}{15565}\right ) \sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )+161564700 \sqrt {3214}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right ) \left (\sqrt {3}+\frac {9 i \sqrt {298434143}}{15565}\right )-7180992 i \sqrt {3}\, \sqrt {298434143}}{200 \left (-\frac {1607}{40}-\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+\frac {\left (-i \sqrt {3}-1\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}}{80}+\frac {1607 i \sqrt {3}}{40}\right ) \left (1773516+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \left (10545+i \left (-10545+\sqrt {298434143}\right ) \sqrt {3}+3 \sqrt {298434143}\right )-276 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+2 i \left (886758+5 \sqrt {298434143}\right ) \sqrt {3}-30 \sqrt {298434143}\right )}&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {4}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}-\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30} & -2 & -7 \\ 0 & \frac {\left (-i \sqrt {895302429}+10545 i \sqrt {3}-3 \sqrt {298434143}-10545\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-1773516 i \sqrt {3}-10 i \sqrt {895302429}+30 \sqrt {298434143}+276 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-1773516}{5 \left (3214+80 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}\right ) \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}} & \frac {19284-360 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+6 i \left (\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214\right ) \sqrt {3}+6 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}}{3214+80 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+i \left (\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214\right ) \sqrt {3}+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = \frac {30 t \left (25 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {3}\, \sqrt {298434143}+3649 i \sqrt {3}\, \sqrt {298434143}\, \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}-2084356 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {3}-845340 i \sqrt {3}\, \sqrt {298434143}+61973045 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \sqrt {3}+75 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {298434143}-10947 \sqrt {298434143}\, \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+2084356 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+61973045 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}-4385905700\right )}{\left (\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )-\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )-40\right ) \left (5 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {3}\, \sqrt {298434143}+299208 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {3}-607 i \sqrt {3}\, \sqrt {298434143}\, \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+15 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {298434143}+5458845 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \sqrt {3}+31140 i \sqrt {3}\, \sqrt {298434143}-299208 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+1821 \sqrt {298434143}\, \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+5458845 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+161564700\right )}, v_{2} = -\frac {10 \left (3 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {895302429}-30 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+505 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {3}-31368 i \sqrt {895302429}+5009248 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \sqrt {3}+9 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {298434143}+90 \sqrt {298434143}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-505 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+5009248 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-162747640\right ) t}{5 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {895302429}+299208 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {3}-607 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {298434143}+5458845 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \sqrt {3}+31140 i \sqrt {895302429}-299208 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+1821 \sqrt {298434143}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+5458845 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+161564700}\right \}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue \(\lambda _{3} = -\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} 3 & -2 & -7 \\ -2 & 0 & 6 \\ 0 & \frac {73}{100} & 2 \end {array}\right ] - \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {4}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}+\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30} & -2 & -7 \\ -2 & -\frac {5}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}+\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30} & 6 \\ 0 & \frac {73}{100} & \frac {1}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}+\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {4}{3}+\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}&-2&-7&0\\ -2&-\frac {5}{3}+\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}&6&0\\ 0&{\frac {73}{100}}&\frac {1}{3}+\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}+\frac {2 R_{1}}{\frac {4}{3}+\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {4}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}+\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}&-2&-7&0\\ 0&\frac {1773516+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \left (10545+i \left (10545+\sqrt {298434143}\right ) \sqrt {3}-3 \sqrt {298434143}\right )-276 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+2 i \left (-886758+5 \sqrt {298434143}\right ) \sqrt {3}+30 \sqrt {298434143}}{5 \left (-3214-80 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214 i \sqrt {3}\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}}&\frac {19284 i+6 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )-19284 \sqrt {3}-360 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{3214 i+\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )-3214 \sqrt {3}+80 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&0\\ 0&{\frac {73}{100}}&\frac {1}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}+\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {73 \left (-3214-80 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214 i \sqrt {3}\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} R_{2}}{20 \left (1773516+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \left (10545+i \left (10545+\sqrt {298434143}\right ) \sqrt {3}-3 \sqrt {298434143}\right )-276 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+2 i \left (-886758+5 \sqrt {298434143}\right ) \sqrt {3}+30 \sqrt {298434143}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {4}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}+\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}&-2&-7&0\\ 0&\frac {1773516+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \left (10545+i \left (10545+\sqrt {298434143}\right ) \sqrt {3}-3 \sqrt {298434143}\right )-276 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+2 i \left (-886758+5 \sqrt {298434143}\right ) \sqrt {3}+30 \sqrt {298434143}}{5 \left (-3214-80 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214 i \sqrt {3}\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}}&\frac {19284 i+6 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )-19284 \sqrt {3}-360 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{3214 i+\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )-3214 \sqrt {3}+80 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}&0\\ 0&0&\frac {38187570+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \left (2217048+\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right ) \left (10545+i \left (10545+\sqrt {298434143}\right ) \sqrt {3}-3 \sqrt {298434143}\right )+3 \left (\left (-\sqrt {298434143}+3515\right ) \sqrt {3}+i \sqrt {298434143}+10545 i\right ) \sqrt {3214}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )+2 i \left (1108524+5 \sqrt {298434143}\right ) \sqrt {3}-30 \sqrt {298434143}\right )+12 \left (-23 \sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )-23 \sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )-3515\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+10 \sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right ) \left (\frac {886758}{5}+i \left (\sqrt {298434143}-\frac {886758}{5}\right ) \sqrt {3}+3 \sqrt {298434143}\right )+30 \sqrt {3214}\, \left (\left (\sqrt {298434143}+\frac {295586}{5}\right ) \sqrt {3}+i \sqrt {298434143}-\frac {886758 i}{5}\right ) \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )+2 i \left (-19093785+2021 \sqrt {298434143}\right ) \sqrt {3}+12126 \sqrt {298434143}}{53205480+30 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \left (10545+i \left (10545+\sqrt {298434143}\right ) \sqrt {3}-3 \sqrt {298434143}\right )-8280 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+60 i \left (-886758+5 \sqrt {298434143}\right ) \sqrt {3}+900 \sqrt {298434143}}&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {4}{3}+\frac {\sqrt {3214}\, \cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30}+\frac {\sqrt {3214}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )}{30} & -2 & -7 \\ 0 & \frac {1773516+\left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \left (10545+i \left (10545+\sqrt {298434143}\right ) \sqrt {3}-3 \sqrt {298434143}\right )-276 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}+2 i \left (-886758+5 \sqrt {298434143}\right ) \sqrt {3}+30 \sqrt {298434143}}{5 \left (-3214-80 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-3214 i \sqrt {3}\right ) \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}} & \frac {19284 i+6 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )-19284 \sqrt {3}-360 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{3214 i+\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )-3214 \sqrt {3}+80 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = \frac {30 t \left (10947 \sqrt {298434143}\, \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+3649 i \sqrt {3}\, \sqrt {298434143}\, \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+25 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {3}\, \sqrt {298434143}-75 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {298434143}+2084356 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {3}+2084356 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-845340 i \sqrt {3}\, \sqrt {298434143}-61973045 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \sqrt {3}+61973045 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}-4385905700\right )}{\left (40+\sqrt {3214}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )+\cos \left (\frac {\arctan \left (\frac {3 \sqrt {895302429}}{15565}\right )}{3}\right )\right )\right ) \left (15 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {298434143}+1821 \sqrt {298434143}\, \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+299208 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {3}+299208 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}}-5 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {2}{3}} \sqrt {3}\, \sqrt {298434143}+5458845 i \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}} \sqrt {3}-5458845 \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}+607 i \sqrt {3}\, \sqrt {298434143}\, \left (31130+6 i \sqrt {3}\, \sqrt {298434143}\right )^{\frac {1}{3}}-161564700-31140 i \sqrt {3}\, \sqrt {298434143}\right )}, v_{2} = -\frac {10 t \left (90 \sqrt {298434143}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+30 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {895302429}+9 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {298434143}+505 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {3}+505 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+31368 i \sqrt {895302429}+5009248 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \sqrt {3}-5009248 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+162747640\right )}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {298434143}+1821 \sqrt {298434143}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+299208 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {3}+299208 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-5 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}} \sqrt {895302429}+5458845 i \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}} \sqrt {3}-5458845 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+607 i \sqrt {895302429}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-161564700-31140 i \sqrt {895302429}}\right \}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] The following table gives a summary of this result. It shows for each eigenvalue the algebraic multiplicity \(m\), and its geometric multiplicity \(k\) and the eigenvectors associated with the eigenvalue. If \(m>k\) then the eigenvalue is defective which means the number of normal linearly independent eigenvectors associated with this eigenvalue (called the geometric multiplicity \(k\)) does not equal the algebraic multiplicity \(m\), and we need to determine an additional \(m-k\) generalized eigenvectors for this eigenvalue.
| multiplicity
| ||||
| eigenvalue | algebraic \(m\) | geometric \(k\) | defective? | eigenvectors |
| \(\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {\frac {1659 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{10}+\frac {2666013}{5 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+16827}{\left (200 \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right )^{2}-\frac {1111 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1785377}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {10583}{3}\right ) \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {4}{3}\right )} \\ -\frac {100 \left (7 \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right )^{2}-\frac {7 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{15}-\frac {22498}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {118}{3}\right )}{200 \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right )^{2}-\frac {1111 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1785377}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {10583}{3}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {-\frac {1659 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{20}-\frac {2666013}{10 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+16827+\frac {4977 i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}}{\left (200 \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {1111 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1785377}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {10583}{3}-\frac {1111 i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {4}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ -\frac {100 \left (7 \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {7 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {11249}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {118}{3}-7 i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )\right )}{200 \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {1111 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1785377}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {10583}{3}-\frac {1111 i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {-\frac {1659 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{20}-\frac {2666013}{10 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+16827-\frac {4977 i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}}{\left (200 \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {1111 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1785377}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {10583}{3}+\frac {1111 i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {4}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ -\frac {100 \left (7 \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {7 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {11249}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {118}{3}+7 i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )\right )}{200 \left (-\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}-\frac {1607}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {1111 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{60}+\frac {1785377}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {10583}{3}+\frac {1111 i \sqrt {3}\, \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\) |
Now that we found the eigenvalues and associated eigenvectors, we will go over each eigenvalue and generate the solution basis. The only problem we need to take care of is if the eigenvalue is defective. Therefore the final solution is \begin {align*} \vec {x}_h(t) &= c_{1} \vec {x}_{1}(t) + c_{2} \vec {x}_{2}(t) + c_{3} \vec {x}_{3}(t) \end {align*}
Which is written as \begin {align*} \left [\begin {array}{c} p \left (t \right ) \\ q \left (t \right ) \\ r \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} \frac {711 \,{\mathrm e}^{\left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right ) t} \left (\frac {7 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {11249}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {71}{3}\right )}{\left (200 \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right )^{2}-\frac {1111 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1785377}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {10583}{3}\right ) \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {4}{3}\right )} \\ -\frac {100 \,{\mathrm e}^{\left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right ) t} \left (7 \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right )^{2}-\frac {7 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{15}-\frac {22498}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {118}{3}\right )}{200 \left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right )^{2}-\frac {1111 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}-\frac {1785377}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}-\frac {10583}{3}} \\ {\mathrm e}^{\left (\frac {\left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}{30}+\frac {1607}{15 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}+\frac {5}{3}\right ) t} \end {array}\right ] + c_{2} \text {Expression too large to display} + c_{3} \text {Expression too large to display} \end {align*}
Which becomes \begin {align*} \text {Expression too large to display} \end {align*}
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 1006
dsolve([diff(p(t),t)=3*p(t)-2*q(t)-7*r(t),diff(q(t),t)=-2*p(t)+6*r(t),diff(r(t),t)=73/100*q(t)+2*r(t)],singsol=all)
\begin{align*} p \left (t \right ) &= -\frac {\left (-i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}+\left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}+96420 i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+9395896 i \sqrt {3}+10228 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-180 i \sqrt {895302429}+540 \sqrt {298434143}-96420 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+9395896\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}+\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-100 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}}}{4800 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}}+\frac {\left (-i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}-\left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}+96420 i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+9395896 i \sqrt {3}-10228 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+180 i \sqrt {895302429}+540 \sqrt {298434143}+96420 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-9395896\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+100 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}}}{4800 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}}+\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}-5114 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-180 i \sqrt {895302429}-96420 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+9395896\right ) c_{3} {\mathrm e}^{\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+50 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right ) t}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}}}{2400 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}} \\ q \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}+\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-100 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+100 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}}+c_{3} {\mathrm e}^{\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+50 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right ) t}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}} \\ r \left (t \right ) &= \frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}-\left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}+32140 i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-10641096 i \sqrt {3}-6228 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-60 i \sqrt {895302429}+180 \sqrt {298434143}-32140 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-10641096\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}+\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-100 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}}}{14400 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}}-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}+\left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}+32140 i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-10641096 i \sqrt {3}+6228 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+60 i \sqrt {895302429}+180 \sqrt {298434143}+32140 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10641096\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}-3214 i \sqrt {3}-\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+100 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}-3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}}}{14400 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}}+\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {4}{3}}-3114 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+60 i \sqrt {895302429}+32140 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+10641096\right ) c_{3} {\mathrm e}^{\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}+50 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}+3214\right ) t}{30 \left (31130+6 i \sqrt {895302429}\right )^{\frac {1}{3}}}}}{7200 \left (31130+6 i \sqrt {895302429}\right )^{\frac {2}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.051 (sec). Leaf size: 602
DSolve[{p'[t]==3*p[t]-2*q[t]-7*r[t],q'[t]==-2*p[t]+6*r[t],r'[t]==73/100*q[t]+2*r[t]},{p[t],q[t],r[t]},t,IncludeSingularSolutions -> True]
\begin{align*} p(t)\to -100 c_2 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {2 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}+111 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]-100 c_3 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {7 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}+1200 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{100}}-200 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-43800 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ] \\ q(t)\to -200 c_1 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {\text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-200 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+200 c_3 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {3 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-200 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{100}}-500 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}+60000 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ] \\ r(t)\to -14600 c_1 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+73 c_2 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {\text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-300 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}+10920000\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{100}}-300 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-40000 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ] \\ \end{align*}