If $A$ is a square matrix and $I$ is an identity matrix such that $A^2=A$, then $A(I-2 A)^3+2 A^3$ is equal to:

A

The correct answer is Option (4) → A

$A^2=A$, then any higher power of $A$ is also equal to A. By extension, $A^n=A$ for any positive integer n.

Similarly, because I is the identity matrix, AI = IA = A

$A(I-2 A)^3+2 A^3$

$A(I^3-8A^3-6A+12A^2)+2A^3$ 

Substituting $A^2=A$ and $\$A^3=A\$\; and\; AI\; =\; A$

$=A - 8A - 6A + 12A + 2A$ 

  = A