Show by an example that for A 0, B 0, AB=0. - Vidyadip

Question

Show by an example that for $\text{A}\neq0,\ \text{B}\neq0,\ \text{AB}=0.$

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Answer

Let $\text{A}=\begin{bmatrix}0&1\\0&2\end{bmatrix}\neq0$ and $\text{B}=\begin{bmatrix}-1&1\\0&0\end{bmatrix}\neq0$
$\therefore\ \text{AB}=\begin{bmatrix}0&1\\0&2\end{bmatrix}\begin{bmatrix}-1&1\\0&0\end{bmatrix}$
$=\begin{bmatrix}0&0\\0&0\end{bmatrix}=0$
Hence proved.

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