If A is a square matrix such that \(A^2 = A\) and \(I\) is the identity matrix of same order as A, then the value of \((A-2I)^2-(2A+I)^2+11A\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(3I\)
A matrix with \(A^2 = A\) is called idempotent. Since A commutes with I (\(AI = IA = A\)), each square expands like an ordinary binomial, with \(A^2 = A\) and \(I^2 = I\):
\( (A-2I)^2 = A^2 - 4A + 4I = -3A + 4I \)
\( (2A+I)^2 = 4A^2 + 4A + I = 8A + I \)
So
\( (-3A + 4I) - (8A + I) + 11A \)
\( = (-3 - 8 + 11)A + (4 - 1)I = 3I \)