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

Answer: (D)

A

Answer 2

$A(I - 2A)^3 + 2A^3$ (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 $( I^3= I)$

⇒$A[I - 6IA + 12A - 8A] + 2A$

⇒$A[I - 14A + 12A] + 2A$

⇒$A[I - 2A] + 2A$

⇒$AI - 2A^2 + 2A$

⇒$A - 2A + 2A$

⇒$A$