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Definition of a Determinant

If B is an id...

Question

If B is an idempotent matrix and $A = I - B$ , then

A

A^{2} = A

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B

A^{2} = I

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C

A B = O

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D

B A = O

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Solution

The correct options are
A $A^{2} = A$
C $A B = O$
D $B A = O$
Since, $B$ is an idempotent matrix,
$B^{2} = B$ .....(1)
$\Rightarrow B (I - B) = O$ where $O$ is a square matrix
Given $A = I - B$ .....(2)
$\Rightarrow B A = O$
Multiplying (2) by $B$
$A B = (I - B) B$
$\Rightarrow A B = B - B^{2}$
$\Rightarrow A B = B - B = O$

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