Question

If $AB=O$, then which of the following is always true

• A. $A=O$
• B. $B=O$
• C. $A$ and $B$ need not be zero matrices
• D. $A$ and $B$ are zero matrices

Answer 1

The correct option is C

$A$ and $B$ need not be zero matrices

Let us take $A=\left[\begin{array}{cc}0& 0\\ a& 0\end{array}\right]$ and $B=\left[\begin{array}{cc}0& 0\\ b& 0\end{array}\right]$.
Where $a\ne 0$ and $b\ne 0$
Now, $AB$
$=\left[\begin{array}{cc}0& 0\\ a& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ b& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
But none of $A$ and $B$ are zero matrices. So if $AB$ is zero it is not necessary that either $A=O$ or $B=O$.