Question

Give examples of matrices

A and B such that AB = O but BA ≠ O.

Answer 1

\[\left( iii \right) Let A = \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix} and B = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\]

\[ \therefore AB = O \]

\[\text{and BA} = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix} \]

\[ \Rightarrow BA = \begin{bmatrix}0 + 0 & 1 + 0 \\ 0 + 0 & 0 + 0\end{bmatrix}\]

\[ \Rightarrow BA = \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}\]

Thus, AB = O but BA ≠ O.