CUET Mathematics 2024 - 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 : | PYQs + Solutions

CUET Mathematics 2024

Algebra

Matrices & Determinants

Easy

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 :

\mathrm{I}+\mathrm{A}

\mathrm{I}+2 \mathrm{~A}

\mathrm{I}-\mathrm{A}

\mathrm{A}

✅ Correct Option: 4

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:

We are given that

A^2 = A

, which means

A

is an idempotent matrix.

First, let's find

A^3

using the given property:

A^3 = A^2 \cdot A = A \cdot A = A

We'll first find

(I-2A)^2

(I-2A)^2 = (I-2A)(I-2A) = I^2 - 2A \cdot I - 2I \cdot A + 4A^2

= I - 2A - 2A + 4A^2

= I - 4A + 4A^2

Since

A^2 = A

, we have:

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

Now we can calculate

(I-2A)^3

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

Next, calculate

A(I-2A)^3

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

Since

A^2 = A

, we have:

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

Finally, calculate the complete expression

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

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

Therefore,

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