Give example of matrices: A and B such that AB ≠ BA - Vidyadip

Question

Give example of matrices:
A and B such that AB ≠ BA

✓

Answer

Let $ \text{A}=\begin{bmatrix}\text{a}&0\\0&0\end{bmatrix},\ \text{B}=\begin{bmatrix}0&\text{b}\\0&0\end{bmatrix}$
$\text{A}\text{B}=\begin{bmatrix}\text{a}&0\\0&0\end{bmatrix}\begin{bmatrix}0&\text{b}\\0&0\end{bmatrix}$
$=\begin{bmatrix}0+0&\text{a}\text{b}+0\\0+0&0+0\end{bmatrix}$
$\text{AB}=\begin{bmatrix}0&\text{a}\text{b}\\0&0\end{bmatrix}$
$\text{BA}=\begin{bmatrix}0&\text{b}\\0&0\end{bmatrix}\begin{bmatrix}0&\text{a}\\0&0\end{bmatrix}$
$ =\begin{bmatrix}0+0&0+0\\0+0&0+0\end{bmatrix}$
$ \text{BA}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$
From equation (i) and (ii)
AB ≠ BA
When $ \text{A}=\begin{bmatrix}\text{a}&0\\0&0\end{bmatrix},\ \text{B}=\begin{bmatrix}0&\text{b}\\0&0\end{bmatrix}$

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