A Matrix Equation.

$($a$)$ Show that if a square matrix A satisfies

$A^2-4A+3I=0$

then

$A^{-1}=\frac{1}{3}(4I-A).$

$($b$)$ Verify these relations in the case that

$A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$

Inverse Property. Let

$A=\left[\begin{array}{ccc}1& 0& 1\\ 2& 3& 4\\ -1& 0& -2\end{array}\right]$

Show, by direct calculation, that

$A^{-1}=-\frac{1}{3}(A^2-2A-4I)$