So here is the complete question:
Use the given definition to find $f\left(A\right)$: if $f$ is the polynomial function $f\left(x\right)= a_0+a_1x+a_2x^2+…+a_nx^n$ then for a square matrix $A,f\left(A\right)$ is defined to be $f\left(A\right)=a_0I+a_1A+a_2A^2+…+a_nA^n$
$$f\left(x\right)=x^2-5x+2$$
$$A=\begin{bmatrix} 2 & 0\\ 4 & 5
\\ \end{bmatrix}$$
So I wasn't sure what to do but I assumed I plugged in $A$ according to the definition supplied. Can you please provide me with tips to get started?
Edited to include the correct matrix supplied by the question.
Edit again:
$$2\begin{bmatrix} 1 & 0\\ 0 & 1
\\ \end{bmatrix}-5\begin{bmatrix} 2 & 0\\ 4 & 5
\\ \end{bmatrix}+\begin{bmatrix} 4 & 0\\ 28 & 25
\\ \end{bmatrix}$$
Is this the correct setup?
Best Answer
The characteristic polynomial of $A$ is $(\lambda-2)(\lambda-5)=\lambda^{2}-7\lambda+10$. So $A^{2}=7A-10 I$, which can be used to reduce any polynomial expression in $A$ to a linear polynomial expression. For example, in this case, $$ A^{2}-5A+2I=2A-8I = 2\left[\begin{matrix}2 & 0 \\ 4 & 5\end{matrix}\right]-8 \left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]. $$ This isn't much easier in your case, but it's a good technique for higher-order expressions.