Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade None
Question:
Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, non-zero symmetric matrix. Then which of the following is (are) skew-symmetric:
$Y^3Z^3 - Z^3Y^3
$X^{44} + Y^{44}
$X^3Z^3 - Z^3X^4
$X^{23} + Y^{23}$
Step-by-Step Solution
Key Concept: A matrix is symmetric if $M^T = M$, skew-symmetric if $M^T = -M$; apply transpose to each expression and compare.
For (A): $(y^3z^3 - z^3y^3)^T = (z^3)^T(y^3)^T - (y^3)^T(z^3)^T = -z^3y^3 + y^3z^3 = y^3z^3 - z^3y^3$, confirming symmetry. For (B): $(x^{44} + y^{44})^T = x^{44} + y^{44}$, also symmetric. For (C): $(x^4z^3 - z^3x^4)^T = -(x^4z^3 - z^3x^4)$, which is skew-symmetric. For (D): $(x^{23} + y^{23})^T = -x^{23} - y^{23}$, which is skew-symmetric.
Correct Answer: 3,4