Question

If $A$ and $B$ are square matrices such that $A^{2020}=O$ and $AB=A+B,$ then $\left|B\right|$ is equal to (where, $O$ is a null matrix)

• A. $0$
• B. $1$
• C. $-1$
• D. $4$

Answer 1

Answer: (A)

$A^{2020}=O\Rightarrow \left|A\right|=0$
$\Rightarrow B=A\left(B-I\right)$
Taking determinant on both sides, we get,
$\left|B\right|=\left|A\right|\left|B-I\right|\Rightarrow \left|B\right|=0$ (as $\left|A\right|=0$ )