Question

Let $A$ and $B$ be $3\times 3$ matrices, then $AB=O$ implies

• A. $A=O$ and $B=O$
• B. $\left|A\right|=0$ and $\left|B\right|=0$
• C. Either $\left|A\right|=0$ or $\left|B\right|=0$
• D. $A=O$ or $B=O$

Answer 1

The correct option is C Either $\left|A\right|=0$ or $\left|B\right|=0$

Given, $A$ and $B$ are $3\times 3$ matrices, then
and $AB=O$
Taking determinant on both sides. we get,
$\Rightarrow |AB|=|O|$
$\Rightarrow |A||B|=0$
$\Rightarrow$ Either $|A|=0$ or $|B|=0$
Option $C$ is correct.