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;

$3I$

The correct answer is Option (3) → $3I$

Given: A is a square matrix such that A² = A (i.e., A is an idempotent matrix), and I is the identity matrix.

We are asked to find the value of:

$\left(A - 2I\right)^2 - \left(2A + I\right)^2 + 11A$

Step 1: Expand both squares using the identity $(X)^2 = X \cdot X$

$\left(A - 2I\right)^2 = A^2 - 4AI + 4I^2 = A - 4A + 4I = -3A + 4I$

$\left(2A + I\right)^2 = 4A^2 + 4AI + I^2 = 4A + 4A + I = 8A + I$

Step 2: Substitute into the expression:

$(-3A + 4I) - (8A + I) + 11A$

= $-3A + 4I - 8A - I + 11A$

= $(-3A - 8A + 11A) + (4I - I)$

= $0A + 3I = 3I$