Question:
If the matrix $A B$ is zero, then
(a) It is not necessary that either $A=O$ or, $B=0$
(b) $A=O$ or $B=0$
(c) $A=O$ and $B=O$
(d) all the above statements are wrong
Solution:
(a) It is not necessary that either $A=O$ or, $B=0$
Let $A=\left[\begin{array}{ll}0 & 2 \\ 0 & 0\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]$
$\therefore A B=\left[\begin{array}{ll}0 & 2 \\ 0 & 0\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
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