Question
If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}1 & -1 & 1 \\ -3 & 2 & -1 \\ -2 & 1 & 0\end{array}\right]$, show that $A B$ and $B A$ are both singular
matrices.
matrices.
$\begin{aligned} & =\left[\begin{array}{ccc}1-6-6 & -1+4+3 & 1-2+0 \\ 2-12-12 & -2+8+6 & 2-4+0 \\ 1-6-6 & -1+4+3 & 1-2+0\end{array}\right] \\ & =\left[\begin{array}{ccc}-11 & 6 & -1 \\ -22 & 12 & -2 \\ -11 & 6 & -1\end{array}\right]\end{aligned}$
$|A B|=\left|\begin{array}{ccc}-11 & 6 & -1 \\ -22 & 12 & -2 \\ -11 & 6 & -1\end{array}\right|$
$=0 \quad \ldots\left[\because R_1\right.$ an $R_3$ are identical $]$
$\mathrm{AB}$ is a singular matrix
$\begin{aligned} \mathbf{B A} & =\left[\begin{array}{ccc}1 & -1 & 1 \\ -3 & 2 & -1 \\ -2 & 1 & 0\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 2 & 3\end{array}\right] \\ & =\left[\begin{array}{ccc}1-2+1 & 2-4+2 & 3-6+3 \\ -3+4-1 & -6+8-2 & -9+12-3 \\ -2+2+0 & -4+4+0 & -6+6+0\end{array}\right] \\ & =\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]\end{aligned}$
$|B A|=0$
BA is a singular matrix.
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