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:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A
Since \(A^2 = A\):
\(A^3 = A^2 \cdot A = A \cdot A = A\)
A and I commute (AI = IA = A), so the usual expansion of a square holds:
\((I - 2A)^2 = I - 4A + 4A^2\)
\(= I - 4A + 4A = I\)
Hence
\((I - 2A)^3 = (I - 2A)^2(I - 2A) = I - 2A\)
\(A(I - 2A)^3 = A - 2A^2 = A - 2A = -A\)
Adding \(2A^3 = 2A\):
\(A(I - 2A)^3 + 2A^3 = -A + 2A = A\)