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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.

In one line: Symmetric mirrors across the diagonal; skew-symmetric mirrors with opposite signs and zero diagonal.

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Understand it

Only square matrices can be tested for symmetry or skew-symmetry. In a skew-symmetric matrix, every diagonal entry must be 0 because aii = -aii gives 2aii = 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.

Example

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

Formula and rule

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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let A = [[1, 2], [3, 4]].

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

Method you can reuse

  1. 1Confirm the matrix is square.
  2. 2Find A'.
  3. 3Compare A' with A for symmetry.
  4. 4Compare A' with -A for skew-symmetry.
  5. 5For decomposition, compute A + A' and A - A', then multiply each by 1/2.

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

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

Common wrong answer

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

Don't mix this up

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

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

  3. 3

    Add for 3 marks

    Only square matrices can be tested for symmetry or skew-symmetry.

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