Question

Let $A$ and $B$ be two non-null square matrices. If the product $AB$ is a null matrix, then

• A. $A$ is singular
• B. $B$ is singular
• C. $A$ is non-singular
• D. $B$ is non-singular

Answer 1

Answer: (A), (B)

$A$ is singular
$B$ is singular

Let $B$ be non-singular, then $B^{-1}$ exists.

Now, $AB=0($ given$)\Rightarrow \left(AB\right)B^{-1}=0B^{-1}$

$($ post multiplying both sides by $B^{-1})$

$\Rightarrow A\left(B{B}^{-1}\right)=0$ $($ by associativity $)$

$\Rightarrow A{I}_n=0(\because B{B}^{-1}=I_n)$

$\Rightarrow A=0$

But $A$ is a non-null matrix.

Hence $B$ is a singular matrix.

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