Question
If $A=\left[\begin{array}{cc}1 & -3 \\ 4 & 2\end{array}\right] B=\left[\begin{array}{cc}4 & 1 \\ 3 & -2\end{array}\right]$, show that $A B \neq B A$.
$=\left[\begin{array}{cc}4-9 & 1+6 \\ 16+6 & 4-4\end{array}\right]$
$=\left[\begin{array}{cc}-5 & 7 \\ 22 & 0\end{array}\right]$
...(i)
$\mathrm{BA}=\left[\begin{array}{rr}4 & 1 \\ 3 & -2\end{array}\right]\left[\begin{array}{cc}1 & -3 \\ 4 & 2\end{array}\right]$
$\begin{aligned} & =\left[\begin{array}{cc}4+4 & -12+2 \\ 3-8 & -9-4\end{array}\right] \\ & =\left[\begin{array}{cc}8 & -10 \\ -5 & -13\end{array}\right]\end{aligned}$
...(ii)
From (i) and (ii), we get AB ≠ BA
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