Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
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: For skew-symmetric matrices A: A^T = -A, and odd powers of skew-symmetric matrices are skew-symmetric while even powers are symmetric. For symmetric matrices B: B^T = B for all powers. A commutator [M,N] = MN - NM is always skew-symmetric.
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