Question

If $A=\left[\begin{array}{c}-2\\ 4\\ 5\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 3& -6\end{array}\right]$, then verify that $(AB{)}^T=B^T{A}^T$.

Answer 1

Now, $AB=\left[\begin{array}{c}-2\\ 4\\ 5\end{array}\right]\left[\begin{array}{ccc}1& 3& -6\end{array}\right]=\left[\begin{array}{ccc}-2& -6& 12\\ 4& 12& -24\\ 5& 15& -30\end{array}\right]$
Thus, $(AB{)}^T=\left[\begin{array}{ccc}-2& 4& 5\\ -6& 12& 15\\ 12& -24& -30\end{array}\right]$ ............ (1)
Now, $B^T{A}^T=\left[\begin{array}{c}1\\ 3\\ -6\end{array}\right]\left[\begin{array}{ccc}-2& 4& 5\end{array}\right]$
$=\left[\begin{array}{ccc}-2& 4& 5\\ -6& 12& 15\\ 12& -24& -30\end{array}\right]$ ............ (2)
From (1) and (2), we get $(AB{)}^T=B^T{A}^T$.