Logo
Jeeify
MCQMediumJEE 2024Algebra of Matrices

JEE Mathematics 2024 Question with Solution

Let AA be a square matrix such that AAT=IAA^T = I. Then 12A[(A+AT)2+(A−AT)2]\frac{1}{2}A[(A + A^T)^2 + (A - A^T)^2] is equal to:

  • A

    A2+IA^2 + I

  • B

    A3+IA^3 + I

  • C

    A2+ATA^2 + A^T

  • D

    A3+ATA^3 + A^T

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: AAT=IAA^T = I, so AA is an orthogonal matrix and hence A−1=ATA^{-1} = A^T.

Find: The value of

12A[(A+AT)2+(A−AT)2]\frac{1}{2}A\left[(A + A^T)^2 + (A - A^T)^2\right]

Expand both squares:

(A+AT)2=A2+AAT+ATA+(AT)2(A + A^T)^2 = A^2 + AA^T + A^TA + (A^T)^2

and

(A−AT)2=A2−AAT−ATA+(AT)2(A - A^T)^2 = A^2 - AA^T - A^TA + (A^T)^2

Adding them,

(A+AT)2+(A−AT)2=2A2+2(AT)2(A + A^T)^2 + (A - A^T)^2 = 2A^2 + 2(A^T)^2

Therefore,

12A[(A+AT)2+(A−AT)2]=A[A2+(AT)2]\frac{1}{2}A\left[(A + A^T)^2 + (A - A^T)^2\right] = A\left[A^2 + (A^T)^2\right]

Now simplify:

A[A2+(AT)2]=A3+A(AT)2A\left[A^2 + (A^T)^2\right] = A^3 + A(A^T)^2

Using AAT=IAA^T = I,

A(AT)2=(AAT)AT=IAT=ATA(A^T)^2 = (AA^T)A^T = IA^T = A^T

Hence,

A[A2+(AT)2]=A3+ATA\left[A^2 + (A^T)^2\right] = A^3 + A^T

Therefore, the correct option is D, that is A3+ATA^3 + A^T.

Identity-Based Simplification

Given: AAT=IAA^T = I.

Use the identity

(X+Y)2+(X−Y)2=2(X2+Y2)(X+Y)^2 + (X-Y)^2 = 2(X^2 + Y^2)

with X=AX = A and Y=ATY = A^T.

So,

(A+AT)2+(A−AT)2=2(A2+(AT)2)(A + A^T)^2 + (A - A^T)^2 = 2\left(A^2 + (A^T)^2\right)

Multiplying by 12A\frac{1}{2}A gives

A(A2+(AT)2)=A3+A(AT)2A\left(A^2 + (A^T)^2\right) = A^3 + A(A^T)^2

Now,

A(AT)2=(AAT)AT=IAT=ATA(A^T)^2 = (AA^T)A^T = IA^T = A^T

Therefore the expression becomes

A3+ATA^3 + A^T

So the correct option is D.

Common mistakes

  • Expanding (A−AT)2(A-A^T)^2 incorrectly by missing the negative middle terms. This is wrong because matrix multiplication still follows the algebraic square expansion. Write it as A2−AAT−ATA+(AT)2A^2 - AA^T - A^TA + (A^T)^2.

  • Assuming (AT)2=A2(A^T)^2 = A^2 for an orthogonal matrix. This is not generally true. Instead, use A(AT)2=(AAT)AT=ATA(A^T)^2 = (AA^T)A^T = A^T based on AAT=IAA^T = I.

  • Using only AAT=IAA^T = I but forgetting that for an orthogonal matrix also ATA=IA^TA = I. This can lead to incomplete simplification when combining terms. Use both identities while expanding the squares.

Practice more Algebra of Matrices questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trialMore on Algebra of Matrices

Related questions