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:
Answer & explanation
Correct answer: option 4
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