Question

$A$ and $B$ be $3\times 3$ matrices. Then $AB=0$ implies :

• A. $A=0$ and $B=0$
• B. $|A|=0$ and $|B|=0$
• C. Either $|A|=0$ or $|B|=0$
• D. $A=0$ or $B=0$

Answer 1

Answer: (C)

Either $|A|=0$ or $|B|=0$

$AB=0\Rightarrow |AB|=0$
$\Rightarrow |A||B|=0\Rightarrow |A|=0$ or $|B|=0$
let $A=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$
Then $AB=0$ but $A\ne 0,B\ne 0$