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

Question

If B is an idempotent matrix, and A = I - B, then (A) A2 = A (B) A2 = I (C) AB = I (D) BA = I

Tardigrade Admin · Accepted Answer

Since B is an idempotent matrix, ∴ B2 = B. Now, A2 = (I - B)2 = (I - B)(I - B) = I - IB- BI + B2 = I - B-B+ B2 = I - 2B+ B2 =I-2B+B=I-B=A ∴ A is idempotent.

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

$A^{2} = A$

$A^{2} = I$

$A B = I$

$B A = I$

Since B is an idempotent matrix, $∴ B^{2} = B .$
Now, $A^{2} = (I - B)^{2} = (I - B) (I - B)$
$= I - I B - B I + B^{2} = I - B - B + B^{2} = I - 2 B + B^{2}$
$= I - 2 B + B = I - B = A$
$∴ A$ is idempotent.

Q. If B is an idempotent matrix, and A=I−B, then

A2=A

A2=I

AB=I

BA=I