Question

If 2 is one of the eigenvalue of





−2 2 −3

2 1 −6

−1 −2 0



 then find the other two eigenvalues.

Answer 1

$A=\begin{pmatrix} -2&2&-3 \\ 2&1&-6\\ -1&-2&0 \end{pmatrix}$

$|A-I\lambda|=\begin{vmatrix} -2-\lambda& 2& -3\\ 2& 1-\lambda & -6\\ -1 & -2 & -\lambda \end{vmatrix}$

$=-(\lambda-5)(\lambda+3)^2$

Equate to zero.

$\implies \lambda=5, -3$