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Question

Mathematics Question on Matrices

If A is a square matrix such that A2=AA^2=A, then (1−A)3+A(1-A)^3+A is equal to

A

AA

B

I−AI - A

C

II

D

3A3A

Answer

II

Explanation

Solution

A is a square matrix such that A2A^2 = A Now (I−A)3+A=(I−A)2(I−A)+A(I - A)^3 + A = (I- A)^2 (I - A) + A = (I2−2AI+A2)(I−A)+A(I^2 - 2AI + A^2) (I - A) + A = (I−2A+A)(I−A)+A(∵ A2=A)(I-2A + A) (I -A) + A (\because \, A^2 = A) = (I−A)(I=A)+A (I -A) (I=A)+A = (I2−2AI+A2)+A(I^2 - 2AI + A^2) + A = (I−2A+A)+A(I - 2A + A) + A (∵ A2=A) (\because \, A^2 =A) = I−A+AI - A + A = 1 ∴ (I−A)3+A=I\therefore \, (I - A)^3 + A = I