Question

If $A=\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right],$ then find $AB,BA.$ Show that $AB\ne BA$.

Answer 1

Given : $A=\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]$
$\therefore AB={\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]}_{2\times 3}{\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]}_{2\times 2}$
$\Rightarrow AB=\left[\begin{array}{cc}2-8+6& 3-10+3\\ -8+8+10& -12+10+5\end{array}\right]$
$\Rightarrow AB=\left[\begin{array}{cc}0& -4\\ 10& 3\end{array}\right]\cdots \left(1\right)$

Now, $BA={\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]}_{3\times 2}\times {\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]}_{2\times 3}$
$\Rightarrow BA=\left[\begin{array}{ccc}2-12& -4+6& 6+15\\ 4-20& -8+10& 12+25\\ 2-4& -4+2& 6+5\end{array}\right]$
$\Rightarrow BA=\left[\begin{array}{ccc}-10& 2& 21\\ -16& 2& 37\\ -2& -2& 11\end{array}\right]....\left(2\right)$
From equation $\left(1\right)$ and $\left(2\right)$
$AB\ne BA$