Skew Symmetric Matrix

Definition: A square matrix A is skew symmetric if: Aᵀ = −A (i.e., the transpose of A equals the negative of A) Equivalently: aᵢⱼ = −aⱼᵢ for all i and j

Key Properties:

1. All diagonal elements of a skew symmetric matrix = 0 (because aᵢᵢ = −aᵢᵢ → 2aᵢᵢ = 0 → aᵢᵢ = 0)
2. Aᵀ = −A
3. A + Aᵀ = 0 (null matrix)
4. If A is skew symmetric, then −A is also skew symmetric
5. Sum of two skew symmetric matrices is skew symmetric
6. If A is skew symmetric, kA is also skew symmetric (k = scalar)
7. Determinant of an odd-order skew symmetric matrix = 0

Example 1 — 2×2 Skew Symmetric Matrix: A = [ 0 3 ] [ -3 0 ]

Aᵀ = [ 0 -3 ] = −A ✓ [ 3 0 ]

Example 2 — 3×3 Skew Symmetric Matrix: A = [ 0 2 -5 ] [ -2 0 3 ] [ 5 -3 0 ]

Verify: Diagonal elements = 0 ✓ aᵢⱼ = −aⱼᵢ: a₁₂ = 2, a₂₁ = -2 → a₁₂ = −a₂₁ ✓

Expressing Matrix as Sum of Symmetric and Skew Symmetric: Every square matrix A can be expressed as: A = ½(A + Aᵀ) + ½(A − Aᵀ) = P + Q Where: • P = ½(A + Aᵀ) → Symmetric matrix • Q = ½(A − Aᵀ) → Skew Symmetric matrix

Example: Let A = [ 1 2 ] [ 3 4 ] Aᵀ = [ 1 3 ] [ 2 4 ] P = ½(A + Aᵀ) = ½[ 2 5 ] = [ 1 2.5 ] [ 5 8 ] [ 2.5 4 ] Q = ½(A − Aᵀ) = ½[ 0 -1 ] = [ 0 -0.5 ] [ 1 0 ] [ 0.5 0 ]

Comparison:

NCERT Class 12 Maths, Chapter 3 — Matrices