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

Hint:

When simplifying matrix expressions, always take into account any given properties of the matrices. In this case, we used \( A^2 = A \) (which means \( A \) is idempotent) to simplify the expression. Also, when expanding expressions like \( (I - 2A)^3 \), be sure to apply the binomial theorem carefully and simplify terms step by step. Always check for matrix properties such as \( I^3 = I \) to help streamline the process.

Answer 1

The Correct Option is D

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

⇒\( A(I - 2A)^3 + 2A \)

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

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

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

⇒\( A[I - 6IA + 12A - 8A] + 2A \)

⇒\( A[I - 14A + 12A] + 2A \)

⇒\( A[I - 2A] + 2A \)

⇒\( AI - 2A^2 + 2A \)

⇒\( A - 2A + 2A \)

⇒\( A \)