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Properties of Determinants

Let matrix A ...

Question

Let matrix $A$ be an idempotent matrix and $B$ be a matrix such that $A + B = l .$
Statement-1 : B is an idempotent matrix. and
Statement-2 : $A B = B A = 0.$

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

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Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

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Statement-1 is True, Statement-2 is False

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Statement-1 is False, Statement-2 is True

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Solution

The correct option is B Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
$A^{2} = A$
$A + B = l \Rightarrow A^{2} + A B = A l \Rightarrow A + A B = A$
$\Rightarrow A B = B A = 0$
$A B + B^{2} = B \Rightarrow B^{2} = B$

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