Question

Let $A$ and $B$ be two non-zero $3 \times 3$ matrices such that $A B=O$. Then

• A. Both $A$ and $B$ are non-singular
• B. Exactly one of $A, B$ is singular
• C. Both $A$ and $B$ are singular
• D. Both $A+B$ and $A B$ are singular

Answer 1

Answer: (C)

Suppose $A$ is non-singular, then
$A^{-1}(A B)=A^{-1}(O) \Rightarrow B=O$
A contradiction.
$\therefore A$ is singular
Similarly, $B$ is singular.