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  4. If A and B are square matrices such that A2020=O and AB=A+B, then |B| is equal to (where, O is a null matrix)

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Q. 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)

NTA AbhyasNTA Abhyas 2020Matrices

Solution:

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