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