Let

$A=([1,2],[-2,1])$

$($a$)$ Show that $A^2-2A+5I=0,$ where I is the identity matrix.

$5$

$($b$)$ Show that $A^{-1}=\frac{1}{5}(2I-A).$

$5$

$($c$)$ Now let $xin{M}_{nn}(R)$ be an arbitrary square matrix $($for some ninN $).$ Show that in case the equation $x^2-2x+5I=0$ holds, $x$ is invertible with inverse $x^{-1}=\frac{1}{5}(2I-x)$

$($d$)$ Let $C$ and $D$ be two arbitrary real $nn$ matrices. Prove that if the matrix $I-CD$ is an invertible matrix, then so is $I-DC.$

$[10$