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Mathematics Question on Matrices

If AA is a square matrix such that A2A^2 = AA , then (I−A)3+A(I-A)^3+A is equal to

A

AA

B

I−AI-A

C

II

D

3A3A

Answer

II

Explanation

Solution

Given that, A2=AA^{2}=A
Then, (I−A)3+A(I-A)^{3}+A
=(I)3+(−A)3+3(I)(−A)2+3(−A)(I)2+A=\left(I\right)^{3}+\left(-A\right)^{3}+3\left(I\right)\left(-A\right)^{2}+3\left(-A\right)\left(I\right)^{2}+A
=I−(A)3+3A2+3(−A)(I)2+A=I-\left(A\right)^{3}+3A^{2}+3\left(-A\right)\left(I\right)^{2}+A
=I−(A)3+3A2−3A+A=I-\left(A\right)^{3}+3A^{2}-3A+A
∵I3=I,IA=A\because I^{3}=I, IA=A
=I−A(A)2+3(A)2−3A+A=I-A\left(A\right)^{2}+3\left(A\right)^{2}-3A+A
=I−A(A)+3A−3A+A=I-A\left(A\right)+3A-3A+A (∵A2=A)\left(\because A^{2}=A\right)
=I−A2+A=I-A^{2}+A
=(I−A)+A(∵A2=A)=\left(I-A\right)+A \left(\because A^{2}=A\right)
=I=I