If $A$ is a square matrix such that $A^2=A$ and $I$ is the identity matrix of the same order as $A$, then $(I+2A)^2 - 5A$ is equal to

$I+3A$

The correct answer is Option (2) → $I+3A$ **

Given: $A^{2}=A$

Compute $(I+2A)^{2}$:

$(I+2A)^{2}=I^{2}+4A+4A^{2}$

$=I+4A+4A$ (since $A^{2}=A$)

$=I+8A$

Now subtract $5A$:

$(I+2A)^{2}-5A = (I+8A)-5A$

$=I+3A$

$I+3A$