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

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

Matrices

Solution:

Since B is an idempotent matrix, $\therefore B^2 = B.$
Now, $A^2 = (I - B)^2 = (I - B)(I - B)$
$= I - IB- BI + B^2 = I - B-B+ B^2 = I - 2B+ B^2$
$=I-2B+B=I-B=A$
$\therefore A$ is idempotent.