Question

$A=\left(\begin{array}{cc}5& 2\\ 7& 3\end{array}\right)$ and $B=\left(\begin{array}{cc}2& -1\\ -1& 1\end{array}\right)$ verify that $(AB{)}^T=B^T{A}^T$

Answer 1

$AB=\left(\begin{array}{cc}5& 2\\ 7& 3\end{array}\right)\left(\begin{array}{cc}2& -1\\ -1& 1\end{array}\right)$

$AB=\left(\begin{array}{cc}10-2& -5+2\\ 14-3& -7+3\end{array}\right)$

$AB=\left(\begin{array}{cc}8& -3\\ 11& -4\end{array}\right)$

${\left(AB\right)}^T={\left(\begin{array}{cc}8& -3\\ 11& -4\end{array}\right)}^T$ .....$\left(1\right)$

$=\left(\begin{array}{cc}8& 11\\ -3& -4\end{array}\right)$

$B^T{A}^T=\left(\begin{array}{cc}2& -1\\ -1& 1\end{array}\right)\left(\begin{array}{cc}5& 7\\ 2& 3\end{array}\right)$

$=\left(\begin{array}{cc}10-2& 14-3\\ -5+2& -7+3\end{array}\right)$

$=\left(\begin{array}{cc}8& 11\\ -3& -4\end{array}\right)$ ....$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we have ${\left(AB\right)}^T=B^T{A}^T$

Hence, verified.