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If $A = \begin{bmatrix} \ -1 & 2 & 3 \\ \ 0 & 3 & 5 \\ \ 0 & 0 & -2\\ \end{bmatrix}$. Find the eigen values of $A^3 +5A +8I$.
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First we find the eigen values of A

To find eigen values of A, characteristic eqation is , |A- λI| =0 , where λ is the eigen value of A & I is the identity matrix

i.e. $\begin{vmatrix} -1-λ&2&3 \\ 0&3-λ&5 \\ 0&0&-2-λ \end{vmatrix} = 0$

Expanding we get a cubic equation in λ as $λ^3 -7λ - 6 =0$ i.e. λ = -1, 3, -2

If λ is the eigen value of A ,then by the property eigen values of $A^3 +5A +8I$ will be $λ^3 +5λ + 8$ ,

By putting λ = -1, 3, -2 ,Eigen values of $A^3 +5A +8I$ will be 2, 50, -10.