Question

If $A=$ $\left[\begin{array}{cc}1& 3\\ -3& 2\end{array}\right]$ and $B=\left[\begin{array}{cc}-2& 1\\ -3& 2\end{array}\right]$. Then find $A^2-5B^2$.

Answer 1

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

$A^2=\left[\begin{array}{cc}1& 3\\ -3& 2\end{array}\right]\left[\begin{array}{cc}1& 3\\ -3& 2\end{array}\right].\left[\begin{array}{cc}1-9& 3+6\\ -3-6& -9+4\end{array}\right]$ $=\left[\begin{array}{cc}-8& 9\\ -9& -5\end{array}\right]$

$B^2=\left[\begin{array}{cc}-2& 1\\ -3& 2\end{array}\right]$ $\left[\begin{array}{cc}-2& 1\\ -3& 2\end{array}\right]$. $\left[\begin{array}{cc}4-3& -2+2\\ 6-6& -3+4\end{array}\right]$

$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$A^2-5B^2=\left[\begin{array}{cc}-8& 9\\ -9& -5\end{array}\right]-5\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$=\left[\begin{array}{cc}-8& 9\\ -9& -5\end{array}\right]-\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$

$=\left[\begin{array}{cc}-13& 9\\ -9& -10\end{array}\right]$.