Question

If $A=\left[\begin{array}{ccc}2& 4& -1\\ -1& 0& 2\end{array}\right],B=\left[\begin{array}{cc}3& 4\\ -1& 2\\ 2& 1\end{array}\right]$, find $(AB{)}^T$.

Answer 1

We have $A=\left[\begin{array}{ccc}2& 4& -1\\ -1& 0& 2\end{array}\right],B=\left[\begin{array}{cc}3& 4\\ -1& 2\\ 2& 1\end{array}\right]$

Since order of $A$ is $2\times 3$ and $B$ is $3\times 2$

$\therefore$ order of $AB=2\times 2$

Hence $AB=\left[\begin{array}{ccc}2& 4& -1\\ -1& 0& 2\end{array}\right]\left[\begin{array}{cc}3& 4\\ -1& 2\\ 2& 1\end{array}\right]$

$=\left[\begin{array}{cc}6-4-2& 8+8-1\\ -3+0+4& -4+0+2\end{array}\right]$

$=\left[\begin{array}{cc}0& 15\\ 1& -2\end{array}\right]$

${\left(AB\right)}^T=\left[\begin{array}{cc}0& 1\\ 15& -2\end{array}\right]$