If A is a square matrix such that $A^2=A

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Matrices

Question:

If A is a square matrix such that $A^2=A,$ then the value of $(I-A)^2-(I+A)^3$ is :

Options:

$-8A$

$8A$

$2(I-4A)$

$2(I+4A)$

Correct Answer:

$-8A$

Explanation:

The correct answer is Option (1) → $-8A$

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

$=(I-A)(I-A)-(I+A)(I+A)(I+A)$

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

$=(I-2A+A)-(I+2A+A)(I+A)$

$=(I-A)-(I+3A)(I+A)$

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

$=I-A-(I+7A)$

$=-8A$