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  4. If A is a square matrix such that A2=A, then (1-A)3+A is equal to

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Q. If A is a square matrix such that $A^2=A$, then $(1-A)^3+A$ is equal to

Matrices

Solution:

A is a square matrix such that $A^2$ = A
Now $(I - A)^3 + A = (I- A)^2 (I - A) + A$
= $(I^2 - 2AI + A^2) (I - A) + A $
= $(I-2A + A) (I -A) + A (\because \, A^2 = A)$
= $ (I -A) (I=A)+A$
= $(I^2 - 2AI + A^2) + A $
= $(I - 2A + A) + A$ $ (\because \, A^2 =A)$
= $I - A + A $= 1 $\therefore \, (I - A)^3 + A = I$