CUET UG 2025 — Mathematics Algebra

A matrix is skew-symmetric if $A^T = -A$AT=−A (the transpose equals the negative of the original matrix).

For example, if $A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$A=[0−2​20​], then $A^T = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} = -A$AT=[02​−20​]=−A

Diagonal elements are always zero, and elements flip signs across the diagonal.

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Let $A$A and $B$B be two skew-symmetric matrices where $A^T = -A$AT=−A and $B^T = -B$BT=−B.

Let $C = A - B$C=A−B

Taking the transpose:

$$C^T = (A - B)^T$$CT=(A−B)T

$$C^T = A^T - B^T$$CT=AT−BT

$$C^T = (-A) - (-B)$$CT=(−A)−(−B)

$$C^T = -A + B$$CT=−A+B

$$C^T = -(A - B)$$CT=−(A−B)

$$C^T = -C$$CT=−C

This shows $C$C is skew-symmetric.

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The difference of two different skew-symmetric matrices is a skew-symmetric matrix.