If A is a square matrix such that $A^2=A$, then $(I−A)^3 + A$ is equal to

The correct answer is $I$

A is a square matrix such that $A^2=A$


$(I−A)^3 +A=(I−A)^2(I−A)+A$

=$(I^2−2AI+A^2)(I−A)+A$

As $A^2 =A$ and $I^2 =I$, the above can be written as

=$(I−2A+A)(I−A)+A$

=$(I−A)(I−A)+A$

=$(I−A)^2+A$

=$I^2−2AI+A^2 +A$

=$I−2A+A+A$


∴ $(I−A)^3 +A=I$