Question

Show that the matrix $A=\left[\begin{array}{cc}5& 3\\ 12& 7\end{array}\right]$ satisfies the equation $A^2-12A-I=0$

Answer 1

$A=\left[\begin{array}{cc}5& 3\\ 12& 7\end{array}\right]$

$A^2=\left[\begin{array}{cc}5& 3\\ 12& 7\end{array}\right]\left[\begin{array}{cc}5& 3\\ 12& 7\end{array}\right]$

$=\left[\begin{array}{cc}25+36& 15+21\\ 60+84& 36+49\end{array}\right]$

$=\left[\begin{array}{cc}61& 36\\ 144& 85\end{array}\right]$ ........$\left(1\right)$

$12A=12\left[\begin{array}{cc}5& 3\\ 12& 7\end{array}\right]=\left[\begin{array}{cc}60& 36\\ 144& 84\end{array}\right]$ ..........$\left(2\right)$

Now $A^2-12A-I$

$=\left[\begin{array}{cc}61& 36\\ 144& 85\end{array}\right]-\left[\begin{array}{cc}60& 36\\ 144& 84\end{array}\right]-\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$=\left[\begin{array}{cc}61-60-1& 36-36-0\\ 144-144-0& 85-84-1\end{array}\right]$

$=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$=0$

Hence proved.