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Question
Give examples of matrices
A and B such that AB = O but A ≠ 0, B ≠ 0.
Solution
\[\left( ii \right) Let A = \begin{bmatrix}0 & 2 \\ 0 & 0\end{bmatrix} and B = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \therefore AB = \begin{bmatrix}0 + 0 & 0 + 0 \\ 0 + 0 & 0 + 0\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} = O\]
Thus, AB = O while A ≠ 0 and B ≠ 0.
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