JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Matrices

Let A and B be any two

3 \times 3

symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Answer: (C)

Solution

Given

A^{T} = A

(symmetric) and

B^{T} = - B

(skew-symmetric).

(A)

C = A^{4} - B^{4}

C^{T} = (A^{4})^{T} - (B^{4})^{T} = A^{4} - B^{4} = C

. Symmetric ✓

(B)

C = A B - B A

C^{T} = (A B)^{T} - (B A)^{T} = B^{T} A^{T} - A^{T} B^{T} = (- B) A - A (- B) = A B - B A = C

. Symmetric ✓

(C)

C = B^{5} - A^{5}

C^{T} = (B^{5})^{T} - (A^{5})^{T} = - B^{5} - A^{5} \neq = - C

. NOT skew-symmetric ✗

(D)

C = A B + B A

C^{T} = (A B)^{T} + (B A)^{T} = - B A - A B = - C

. Skew-symmetric ✓

Option (C) is NOT true.

JEE Main · 2022 · 28 Jul 2022 · Shift 2 · Single correct option