Question

If $A=\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}-2& 1\\ -3& 2\end{array}\right]$ and $5A^2-B^2=5C$. Find matrix C where C is a $2$ by $2$ matrix.

Answer 1

$A=\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]$

$A^2=\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]\left[\begin{array}{cc}1& 3\\ 3& 4\end{array}\right]$

$A^2=\left[\begin{array}{cc}10& 15\\ 15& 25\end{array}\right]$

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

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

$B^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$5A^2-B^2=\left[\begin{array}{cc}49& 75\\ 75& 124\end{array}\right]$

$5C=\left[\begin{array}{cc}49& 75\\ 75& 124\end{array}\right]$

Therefore $C=\left[\begin{array}{cc}\frac{49}{5}& 15\\ 15& \frac{124}{5}\end{array}\right]$