11 min readConcept sheet1-mark MCQ or assertion-reason; 2-mark proofs; 3 or 5-mark 'express as a sum' questions; also a JEE favourite

Symmetric and skew symmetric matrices

A square matrix is symmetric if A′ = A and skew symmetric if A′ = −A. Every square matrix splits into one of each.

SymmetricA′ = A, that is aᵢⱼ = aⱼᵢ: a mirror image across the main diagonal

Skew symmetricA′ = −A, that is aᵢⱼ = −aⱼᵢ. Every diagonal entry is 0.

Building themFor any square A, A + A′ is symmetric and A − A′ is skew symmetric.

The splitA = P + Q, with P = ½(A + A′) symmetric and Q = ½(A − A′) skew symmetric

Mirror across the diagonal

A square matrix A is symmetric if A′ = A: each entry equals its mirror image across the main diagonal, aᵢⱼ = aⱼᵢ. Example: [2, 5, −1; 5, 0, 3; −1, 3, 7]. A is skew symmetric if A′ = −A: each entry is the negative of its mirror image, aᵢⱼ = −aⱼᵢ. Example: [0, 4, −2; −4, 0, 6; 2, −6, 0]. The zero matrix is both. (Matrices are written row by row here, with rows separated by semicolons.)

Proof: the diagonal of a skew symmetric matrix is zero

1
A is skew symmetric, so aᵢⱼ = −aⱼᵢ for all i and j.
2
Put j = i: aᵢᵢ = −aᵢᵢ.
3★
So 2aᵢᵢ = 0.
4
So, aᵢᵢ = 0 for every i: all the diagonal entries are zero.

★ key step

Proof: every square matrix is a sum of a symmetric and a skew symmetric matrix

1
Let P = A + A′. Then P′ = A′ + (A′)′ = A′ + A = P. So A + A′ is symmetric.
2
Let Q = A − A′. Then Q′ = A′ − (A′)′ = A′ − A = −(A − A′) = −Q. So A − A′ is skew symmetric.
3★
Write A = ½(A + A′) + ½(A − A′). The right side adds up to ½A + ½A = A.
4
Multiplying by ½ keeps each type, so ½(A + A′) is symmetric and ½(A − A′) is skew symmetric.
5
So, every square matrix is the sum of a symmetric matrix and a skew symmetric matrix.

★ key step

Example 1: express a 2 × 2 matrix as a sum

Problem

Express A = [4, 7; −1, 2] as the sum of a symmetric and a skew symmetric matrix.

Working

Rule: P = ½(A + A′) is symmetric and Q = ½(A − A′) is skew symmetric.
A′ = [4, −1; 7, 2]
A + A′ = [8, 6; 6, 4], so P = [4, 3; 3, 2]. P′ = P.
A − A′ = [0, 8; −8, 0], so Q = [0, 4; −4, 0]. Q′ = −Q.
Check: P + Q = [4 + 0, 3 + 4; 3 − 4, 2 + 0] = [4, 7; −1, 2] = A.

Answer

A = [4, 3; 3, 2] + [0, 4; −4, 0]

Example 2: a 3 × 3 matrix (5-mark style)

Problem

Express A = [2, 4, −6; 0, 3, 1; 2, −1, 5] as the sum of a symmetric and a skew symmetric matrix.

Working

Rule: A = P + Q with P = ½(A + A′) and Q = ½(A − A′).
A′ = [2, 0, 2; 4, 3, −1; −6, 1, 5]
A + A′ = [4, 4, −4; 4, 6, 0; −4, 0, 10], so P = [2, 2, −2; 2, 3, 0; −2, 0, 5]. P′ = P.
A − A′ = [0, 4, −8; −4, 0, 2; 8, −2, 0], so Q = [0, 2, −4; −2, 0, 1; 4, −1, 0]. Q′ = −Q.
Check row by row: P + Q = [2, 4, −6; 0, 3, 1; 2, −1, 5] = A.

Answer

A = [2, 2, −2; 2, 3, 0; −2, 0, 5] + [0, 2, −4; −2, 0, 1; 4, −1, 0]

Quick facts

If A is symmetric, so is kA for any scalar k, and so is A², since (A²)′ = (A′)² = A².
If A is symmetric, then B′AB is symmetric for any square B of the same order: (B′AB)′ = B′A′(B′)′ = B′AB.
If A and B are symmetric and of the same order, then AB is symmetric exactly when AB = BA.
For any matrix A, both AA′ and A′A are symmetric.

In real life

A road distance chart between towns such as Chennai, Madurai and Coimbatore is a symmetric matrix: Chennai to Madurai equals Madurai to Chennai, and every diagonal entry (a town to itself) is 0.

Exam angle

1 marks: “If A = [0, a, 3; 2, 0, b; c, −1, 0] is skew symmetric, find a, b and c.” aᵢⱼ = −aⱼᵢ gives a = −2, b = −(−1) = 1 and c = −3.

3 marks: “Express [2, 9; −3, 5] as the sum of a symmetric and a skew symmetric matrix.” A′ = [2, −3; 9, 5]. P = ½(A + A′) = [2, 3; 3, 5] and Q = ½(A − A′) = [0, 6; −6, 0]. Check: P + Q = [2, 9; −3, 5].

2 marks: “If A is a symmetric matrix, show that B′AB is symmetric for any square matrix B of the same order.” (B′AB)′ = B′A′(B′)′ by the reversal law. Since A′ = A and (B′)′ = B, this equals B′AB. So B′AB is symmetric.

Remember it

Symmetric: mirror entries equal. Skew: mirror entries opposite, diagonal zero. Split any square A as ½(A + A′) + ½(A − A′).