Question

If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$,then show that $A^2-5A+7I_2=0$

Answer 1

$A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]A^2=5A+7I_2=0$

$A.A=\left[\begin{array}{cc}-3& 1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}-3& 1\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}9-1& +3-2\\ 63+2& -1+4\end{array}\right]=\left[\begin{array}{cc}8& +5\\ 5& 3\end{array}\right]$

$=\left[\begin{array}{cc}8& +5\\ 5& 3\end{array}\right]-\left[\begin{array}{cc}15& 5\\ -5& 10\end{array}\right]+\left[\begin{array}{cc}7& 0\\ 0& 7\end{array}\right]$

$=\left[\begin{array}{cc}15& +5\\ 5& 10\end{array}\right]-\left[\begin{array}{cc}15& 5\\ 5& 10\end{array}\right]=0$