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Symmetric and Skew-Symmetric Matrices

A square matrix A is symmetric if A' = A. It is skew-symmetric if A' = -A.

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Student-friendly explanation

Only square matrices can be tested for symmetry or skew-symmetry. In a skew-symmetric matrix, every diagonal entry must be 0 because a_ii = -a_ii gives 2a_ii = 0. Every square matrix A can be written as the sum of a symmetric matrix and a skew-symmetric matrix: A = (A + A')/2 + (A - A')/2.

How to write this in exams

  1. 1

    Start with the exact idea

    A square matrix A is symmetric if A' = A. It is skew-symmetric if A' = -A.

  2. 2

    Then show how to use it

    Confirm the matrix is square. Find A'. Compare A' with A for symmetry. Compare A' with -A for skew-symmetry. For decomposition, compute A + A' and A - A', then multiply each by 1/2.

  3. 3

    Add one concrete example

    [[2, 5], [5, 3]] is symmetric. [[0, 4], [-4, 0]] is skew-symmetric.

  4. 4

    Avoid this incomplete answer

    Testing only off-diagonal entries for skew-symmetry and ignoring non-zero diagonal entries.

Definition

A square matrix A is symmetric if A' = A. It is skew-symmetric if A' = -A.

Example

[[2, 5], [5, 3]] is symmetric. [[0, 4], [-4, 0]] is skew-symmetric.

Rule to remember

Symmetric condition: A' = A. Skew-symmetric condition: A' = -A. Decomposition: A = 1/2(A + A') + 1/2(A - A'), where 1/2(A + A') is symmetric and 1/2(A - A') is skew-symmetric.

Memory hook

Symmetric mirrors across the diagonal; skew-symmetric mirrors with opposite signs and zero diagonal.

Examples and method

Worked example

Let A = [[1, 2], [3, 4]]. A' = [[1, 3], [2, 4]]. Symmetric part = 1/2(A + A') = 1/2[[2, 5], [5, 8]] = [[1, 5/2], [5/2, 4]]. Skew-symmetric part = 1/2(A - A') = 1/2[[0, -1], [1, 0]] = [[0, -1/2], [1/2, 0]]. Their sum is [[1, 2], [3, 4]] = A.

Method to apply

Confirm the matrix is square. Find A'. Compare A' with A for symmetry. Compare A' with -A for skew-symmetry. For decomposition, compute A + A' and A - A', then multiply each by 1/2.

Diagram support

No diagram is needed; the property is checked by comparing positions across the main diagonal.

How CBSE asks it

Frequently asked as identification, proof of properties, finding unknown entries, or expressing a matrix as a sum of symmetric and skew-symmetric matrices.

Avoid common mistakes

Common confusion

Students often forget that a skew-symmetric matrix must have zero diagonal entries.

Common wrong answer

Testing only off-diagonal entries for skew-symmetry and ignoring non-zero diagonal entries.

Exam tip

For decomposition questions, always write A as half of (A + A') plus half of (A - A'); then identify the first part as symmetric and the second as skew-symmetric.

Quick check

Can [[1, 2], [-2, 0]] be skew-symmetric?

No. Its off-diagonal entries fit the pattern, but the first diagonal entry is 1, not 0.

Answer writing and exam use

1-mark answer

A square matrix A is symmetric if A' = A. It is skew-symmetric if A' = -A.

2-mark answer

A square matrix A is symmetric if A' = A. It is skew-symmetric if A' = -A. Symmetric condition: A' = A. Skew-symmetric condition: A' = -A. Decomposition: A = 1/2(A + A') + 1/2(A - A'), where 1/2(A + A') is symmetric and 1/2(A - A') is skew-symmetric. [[2, 5], [5, 3]] is symmetric. [[0, 4], [-4, 0]] is skew-symmetric.

3-mark answer

Only square matrices can be tested for symmetry or skew-symmetry. In a skew-symmetric matrix, every diagonal entry must be 0 because a_ii = -a_ii gives 2a_ii = 0. Every square matrix A can be written as the sum of a symmetric matrix and a skew-symmetric matrix: A = (A + A')/2 + (A - A')/2. Symmetric condition: A' = A. Skew-symmetric condition: A' = -A. Decomposition: A = 1/2(A + A') + 1/2(A - A'), where 1/2(A + A') is symmetric and 1/2(A - A') is skew-symmetric. Let A = [[1, 2], [3, 4]]. A' = [[1, 3], [2, 4]]. Symmetric part = 1/2(A + A') = 1/2[[2, 5], [5, 8]] = [[1, 5/2], [5/2, 4]]. Skew-symmetric part = 1/2(A - A') = 1/2[[0, -1], [1, 0]] = [[0, -1/2], [1/2, 0]]. Their sum is [[1, 2], [3, 4]] = A. Frequently asked as identification, proof of properties, finding unknown entries, or expressing a matrix as a sum of symmetric and skew-symmetric matrices. Testing only off-diagonal entries for skew-symmetry and ignoring non-zero diagonal entries.
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