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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 = 0

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D

B A = 0

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Solution

The correct options are
A $A^{2} = A$
C $A B = 0$
D $B A = 0$
Property of Idempotent matrix $B^{2} = B$
$A = I - B$

Multiply with B on both sides.
$B A = B I - B^{2}$
$B A = B - B = 0$

Hence $B A = 0 = A B$

$A = I - B \Rightarrow A A = A I - A B \Rightarrow A^{2} = A - A B$
As $A B = 0$
$A^{2} = A - 0 \Rightarrow A^{2} = A$

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