Question

If matrix $A=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$, then prove that $A^2-4A+5I=0$, where $I$ is a unitary matrix.

Answer 1

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

$\Rightarrow A^2=$ $\left[\begin{array}{cc}-1& -4\\ 8& 7\end{array}\right]$

$\Rightarrow A^2-4A=$ $\left[\begin{array}{cc}-5& 0\\ 0& -5\end{array}\right]=-5I\Rightarrow A^2-4A+5I=0$