Question

If $A$ and $B$ are two non-zero square matrices of the same order such that the product $AB=0$, then

• A. both A and B must be singular
• B. exactly one of them must be singular
• C. atleast one of them must be non-singular
• D. none of these

Answer 1

Answer: (A)

both A and B must be singular

Assume that $A$ is non-singular, then $A^{-1}$ exists. Thus
$AB=0\Rightarrow A^{-1}\left(AB\right)=\left(A^{-1}A\right)B=0$
$\Rightarrow IB=0$
$\therefore B=0$. A contradiction.
$\Rightarrow$ A is singular, similarly B is also singular.
Hence, both A and B must be singular.