Question

Give examples of matrices
A and B such that AB ≠ BA

Answer 1

\[\left( i \right) Let A = \begin{bmatrix}1 & - 2 \\ 3 & 2\end{bmatrix} and B = \begin{bmatrix}2 & 3 \\ - 1 & 2\end{bmatrix}\]

\[AB = \begin{bmatrix}1 & - 2 \\ 3 & 2\end{bmatrix}\begin{bmatrix}2 & 3 \\ - 1 & 2\end{bmatrix}\]

\[ \Rightarrow AB = \begin{bmatrix}2 + 2 & 3 - 4 \\ 6 - 2 & 9 + 4\end{bmatrix}\]

\[ \Rightarrow AB = \begin{bmatrix}4 & - 1 \\ 4 & 13\end{bmatrix}\]

\[Now, \]

\[BA = \begin{bmatrix}2 & 3 \\ - 1 & 2\end{bmatrix}\begin{bmatrix}1 & - 2 \\ 3 & 2\end{bmatrix}\]

\[ \Rightarrow BA = \begin{bmatrix}2 + 9 & - 4 + 6 \\ - 1 + 6 & 2 + 4\end{bmatrix}\]

\[ \Rightarrow BA = \begin{bmatrix}11 & 2 \\ 5 & 6\end{bmatrix}\]

\[\]

Thus, AB ≠ BA.