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

Mumbai University > COMPS > Sem 4 > Applied Mathematics 4

Marks : 05

Year : DEC 2014

Since A is a upper triangle matrix, Eigen values $(λ) = $ diagonal elements $=-1, 3,-2$

Now, we know, $F(λ)$ is Eigen value of $F(A)$

Let $f(A)=A^3+5A+8I\\ f(\lambda)=\lambda^3+5\lambda+8$

When $\lambda=-1,f(-1)=(-1)^3+5(-1)+8=2$

When $\lambda=3,f(3)=(3)^3+5(3)+8=50$

When $\lambda=-2,f(-2)=(-2)^3+5(-2)+8=-10$

$\therefore $ Eigen value of $A^3+5A+8I$ are $2,50,-10$