If A is a square matrix such that $A^2=A,$ then the value of $(I-A)^2-(I+A)^3$ is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $-8A$
$A^2=A$ so $(I-A)^2-(I+A)^3$
$=(I-A)(I-A)-(I+A)(I+A)(I+A)$
$=(I-A-A+A^2)-(I+A+A+A^2)(I+A)$
$=(I-2A+A)-(I+2A+A)(I+A)$
$=(I-A)-(I+3A)(I+A)$
$=I-A-(I+A+3A+3A^2)$
$=I-A-(I+7A)$
$=-8A$