If A is a square matrix such that \(A^2=A\) and \(I\) is the identity matrix of the same order as \(A\) then \((I + 2A)^3-6A\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(I + 20A\)
Since \(A^2 = A\) (an idempotent matrix), every higher power is also A:
\(A^3 = A^2 \cdot A = A \cdot A = A\)
I and A commute, so the binomial expansion holds:
\((I + 2A)^3 = I + 3(2A) + 3(2A)^2 + (2A)^3\)
\(= I + 6A + 12A^2 + 8A^3\)
\(= I + 6A + 12A + 8A\)
\(= I + 26A\)
Hence
\((I + 2A)^3 - 6A = I + 26A - 6A = I + 20A\)
Why the other options are incorrect
Option (1) \(I + 26A\) — Incorrect. This is \((I + 2A)^3\) alone; the \(-6A\) has not been subtracted.
Option (2) \(20A\) — Incorrect. The term \(I^3 = I\) of the expansion is missing.
Option (4) \(26A\) — Incorrect. Both the I term and the \(-6A\) are missing.