Question

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

Answer 1

$A=\left[\begin{array}{cc}5& 2\\ 7& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}2& -1\\ -1& 1\end{array}\right]$

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

Let $C=AB$

$C=\left[\begin{array}{cc}5& 2\\ 7& 3\end{array}\right]\left[\begin{array}{cc}2& -1\\ -1& 1\end{array}\right]$

$=\left[\begin{array}{cc}5\times 2+2\times -1& 5\times -1+2\times 1\\ 7\times 2+3\times -1& 7\times -1+3\times 1\end{array}\right]$

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

Let $D=B^T{A}^T$

$D=\left[\begin{array}{cc}2& -1\\ -1& 1\end{array}\right]\left[\begin{array}{cc}5& 7\\ 2& 3\end{array}\right]$

$=\left[\begin{array}{cc}2\times 5+-1\times 2& 2\times 7+-1\times 3\\ -1\times 5+1\times 2& -1\times 7+1\times 3\end{array}\right]$

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

$C^T=D$

which means $(AB{)}^T{=B}^T{A}^T$