Question

If the product of two non-null square matrix is a null matrix, then

• A. only one of them is singular
• B. both of them must be singular
• C. both of them are non singular
• D. none of these

Answer 1

The correct option is B both of them must be singular

Let $A$ & $B$ be two non null matrices of same order $n\times n$ .
Given that $AB=O$
If possible, let $B$ be a non singular matrix, then B${}^{-1}$ exists .
Now $AB=O$
$\Rightarrow \left(AB\right)B^{-1}=O{B}^{-1}$
$\Rightarrow A=O$

But $A$ is non null matrix.

Hence, our assumption is wrong.
So, $B$ is a singular matrix .

Similarly, it can be shown that $A$ is a singular matrix.

Hence, option B.