If A is a square matrix such that \(\mathrm{A}^2 = \mathrm{A}\) and I is the identity matrix of same order as A, then the matrix \((2\mathrm{I} + \mathrm{A})^3 - 19\mathrm{A} - 3\mathrm{I}\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 5 I
Since \(A^2 = A\) (A is idempotent):
\(A^3 = A^2 \cdot A = A \cdot A = A\)
I commutes with A, so the binomial expansion holds:
\((2I + A)^3 = 8I + 3(4I)A + 3(2I)A^2 + A^3\)
\(= 8I + 12A + 6A + A\)
\(= 8I + 19A\)
Hence:
\((2I + A)^3 - 19A - 3I = 8I + 19A - 19A - 3I\)
\(= 5I\)