Finding eigenvalue of a matrix expression

Question

Consider the following matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 2 & 3 & 0 \\ -3 & 1 & -2\end{array}\right]$

How can I find the eigenvalues of $3 \mathrm{~A}^{3}+5 \mathrm{~A}^{2}-6 \mathrm{~A}+2 \mathrm{I}$?

I know the matrix $A$ has eigenvalues $1,3,-2$ and also that I can use cayley Hamilton theorem and reduce the matrix expression, but that doesn't reduce it enough so that I can avoid multiplication of matricies.

Can anyone please help me how to do this?

Thank you.

Answer 1

Hint: If $Av = \lambda v$, then what can we say about $(3A^3 +5A^2 - 6A + 2I)v$?

Answer 2

If $a$ is a eigen value of the matrix $A$ then $a^2$ is the eigen value for $A^2$...similar arguments for $A^m$, where m is a positive integer.

Similarly, $k*a$ is the eigen value for $kA$.

Now use the previously mentioned rules (observations) to calculate the eigen value of any expression with out calculating the powers of the matrix.