Question

Let X and Y be two arbitrary, 3 × 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 × 3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric?

(A) Y³Z⁴ - Z⁴Y³

(B) X⁴⁴ + Y⁴⁴

(C) X⁴Z³ - Z³X⁴

(D) X²³ + Y²³

Answer 1

Answer is (C) (D)

X, Y → skew-symmetric matrices of order 3 × 3

Z → symmetric matrix of order 3 × 3

and X, Y, Z ≠ 0

Checking all the options:

Option (A) (Y³Z⁴ - Z⁴Y³)^T= (Y³Z⁴)^T - (Z⁴Y³)^T

= (Z⁴)^T (Y³)^T - (Y³)^T (Z⁴)^T

= (Z^T) 4(Y^T)³ - (Y^T) 3(Z^T)⁴ = −Z⁴Y³ + Y³Z⁴

Option (B) (X⁴⁴ + Y⁴⁴)^T = (X^T)⁴⁴ + (Y^T)⁴⁴

= X⁴⁴ + Y⁴⁴ ⇒ (symmetric)

Option (C) (X⁴Z³ - Z³X⁴)^T = (Z^T)³ (X^T)⁴ - (X^T)⁴(Z^T)³

= Z³X⁴ - X⁴Z³ ⇒ (skew-symmetric)

Option (D) (X²³ + Y²³)^T = (X^T)²³ + (Y^T)²³

= (−X)²³ + (−Y)²³ = −(X²³ + Y²³)

⇒ (skew-symmetric)