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Skew Hermitian Matrix

A square matr...

Question

A square matrix $A$ is said to be an idempotent matrix if $A^{2} = A$ and $B = I - A$ , then

B

is an idempotent

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A B = I_{n}

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B A^{'} = I_{n}

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none of these

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Solution

The correct option is A $B$ is an idempotent
$A is an idempotent matrix and B=I-A i . e A^{2} = A a n d B = I - A B^{2} = (I - A) (I - A) = I^{2} - I A - A I + A^{2} = I - A - A + A \Rightarrow B^{2} = I - A = B Hence, B is an idempotent.$

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(2 I + A)^{4} = a I + b A, (I is unit matrix), a, b \in R .

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