Question

If $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$ and $B\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$ find $A\left(BA\right)$.

Answer 1

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

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

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

$=\left[\begin{array}{cc}4& 5\\ 5& 4\end{array}\right]$

$A\left(BA\right)=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]\left[\begin{array}{cc}4& 5\\ 5& 4\end{array}\right]$

$=\left[\begin{array}{cc}4+10& 5+8\\ 8+5& 10+4\end{array}\right]$

$=\left[\begin{array}{cc}14& 13\\ 13& 14\end{array}\right]$.