Chapter 2.2, Problem 59E

Polynomial Function In Exercises 59 and 60, find $\mathit{f}\left(\mathit{A}\right)$ using the definition below.

If $\mathit{f}\left(\mathit{x}\right)={\mathit{a}}_0+{\mathit{a}}_1\mathit{x}+{\mathit{a}}_2{\mathit{x}}^2+\dots +{\mathit{a}}_{\mathit{n}}{\mathit{x}}^{\mathit{n}}$ is a polynomial function, then for a square matrix $\mathit{A}$,

$\mathit{f}\left(\mathit{A}\right)={\mathit{a}}_0\mathit{I}+{\mathit{a}}_1\mathit{A}+{\mathit{a}}_2{\mathit{A}}^2+\dots +{\mathit{a}}_{\mathit{n}}{\mathit{A}}^{\mathit{n}}$.

$f\left(x\right)=2-5x+x^2$, $A=\left[\begin{array}{cc}2& 0\\ 4& 5\end{array}\right]$.