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GUJCET 2024 Math Paper (04)

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Question : 18 of 40

Marks: +1, -0

If

A

is square matrix such that

A^2=A

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

$2( I - A )$
I-A
I
0
None of above

Solution:

Concept:

Use the idempotent property

A^2=A

to simplify the matrix expression.

Explanation:

Given

A^2=A

.

First simplify

(I-A)^3

:

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

.

So

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

.

Now simplify

(I+A)^2

:

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

.

Therefore,

(I-A)^3 - (I+A)^2 = (I-A) - (I+3A) = -4A

.

This is not among the given options A to D.

Answer:

-4A

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