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Question
Give examples of matrices
A and B such that AB ≠ BA
Solution
\[\left( i \right) Let A = \begin{bmatrix}1 & - 2 \\ 3 & 2\end{bmatrix} and B = \begin{bmatrix}2 & 3 \\ - 1 & 2\end{bmatrix}\]
\[AB = \begin{bmatrix}1 & - 2 \\ 3 & 2\end{bmatrix}\begin{bmatrix}2 & 3 \\ - 1 & 2\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}2 + 2 & 3 - 4 \\ 6 - 2 & 9 + 4\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}4 & - 1 \\ 4 & 13\end{bmatrix}\]
\[Now, \]
\[BA = \begin{bmatrix}2 & 3 \\ - 1 & 2\end{bmatrix}\begin{bmatrix}1 & - 2 \\ 3 & 2\end{bmatrix}\]
\[ \Rightarrow BA = \begin{bmatrix}2 + 9 & - 4 + 6 \\ - 1 + 6 & 2 + 4\end{bmatrix}\]
\[ \Rightarrow BA = \begin{bmatrix}11 & 2 \\ 5 & 6\end{bmatrix}\]
\[\]
Thus, AB ≠ BA.
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