Question 8.8.3: Find the eigenvalues and eigenvectors of = (3 4 -1 7)....
Find the eigenvalues and eigenvectors of \mathrm{A}=\left(\begin{array}{rr}3 & 4 \\ -1 & 7\end{array}\right).
From the characteristic equation
\operatorname{det}(\mathrm{A}-\lambda \mathrm{I})=\left|\begin{array}{cc} 3-\lambda & 4 \\ -1 & 7-\lambda \end{array}\right|=(\lambda-5)^{2}=0,
we see \lambda_{1}=\lambda_{2}=5 is an eigenvalue of multiplicity 2 . In the case of a 2 \times 2 matrix, there is no need to use Gauss-Jordan elimination. To find the eigenvector(s) corresponding to \lambda_{1}=5, we resort to the system (\mathrm{A}-\mathrm{5 I} \mid \mathrm{0}) in its equivalent form
\begin{aligned} -2 k_{1}+4 k_{2} & =0 \\ -k_{1}+2 k_{2} & =0. \end{aligned}
It is apparent from this system that k_{1}=2 k_{2}. Thus, if we choose k_{2}=1, we find the single eigenvector \mathrm{K}_{1}=\left(\begin{array}{l}2 \\ 1\end{array}\right).