Give example of matrices: A and B such that AB = 0 but BA ≠ 0 - Vidyadip

Question

Give example of matrices:
A and B such that AB = 0 but BA ≠ 0

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Answer

Let $\text{A}=\begin{bmatrix}0&1\\0&0\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0\\0&0\end{bmatrix}$
$ \therefore\ \text{AB}=0$
and $\text{BA}=\begin{bmatrix}1&0\\0&0\end{bmatrix}\begin{bmatrix}0&1\\0&0\end{bmatrix}$
$\Rightarrow\text{BA}=\begin{bmatrix}0+0&1+0\\0+0&0+0\end{bmatrix}$
$\Rightarrow\text{BA}=\begin{bmatrix}0&1\\0&0\end{bmatrix}$
Thus, AB = 0 but BA ≠ 0

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