Question

If \( A \) is a square matrix and \( I \) is an identity matrix such that \( A^2 = A \), then \( A(I - 2A)^3 + 2A^3 \) is equal to:

• A. I+A
• B. I+2A
• C. I−A
• D. A

Answer 1

The Correct Option is D

\( A(I - 2A)^3 + 2A^3 \) (Since \( A^2 = A \))

⇒\( \Rightarrow A(I - 2A)^3 + 2A \)

⇒\( A[I^3 - 3I^2(2A) + 3I(2A)^2 - (2A)^3] + 2A \)

⇒\( A[I^3 - 6I^2A + 12IA^2 - 8A^3] + 2A \)

⇒\( A[I^3 - 6I^2A + 12IA^2 - 8A] + 2A \)   we know that \(( I^3= I)\)

⇒\( A[I - 6IA + 12A - 8A] + 2A \)

⇒\( A[I - 14A + 12A] + 2A \)

⇒\( A[I - 2A] + 2A \)

⇒\( AI - 2A^2 + 2A \)

⇒\( A - 2A + 2A \)

⇒\( A \)