Symmetric and Skew-Symmetric Matrices

The two definitions

A square matrix $A$ is symmetric if $A^T = A$ , and skew-symmetric if $A^T = -A$ . Every square matrix splits into a symmetric part plus a skew-symmetric part:

A = \underbrace{\dfrac{1}{2}(A+A^T)}_{\text{symmetric}} + \underbrace{\dfrac{1}{2}(A-A^T)}_{\text{skew-symmetric}}

Worked example: a 2×2 matrix

Split $A = \begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$ . With $A^T = \begin{bmatrix} 3 & 1 \\ 5 & -1 \end{bmatrix}$ :

\dfrac{1}{2}(A+A^T) = \begin{bmatrix} 3 & 3 \\ 3 & -1 \end{bmatrix}, \qquad \dfrac{1}{2}(A-A^T) = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}

The first equals its transpose, the second's transpose is its negative, and their sum is $A$ .

Worked example: a 3×3 matrix

Split $A = \begin{bmatrix} 1 & 2 & 4 \\ 6 & 8 & 1 \\ 3 & 5 & 7 \end{bmatrix}$ .

\dfrac{1}{2}(A+A^T) = \begin{bmatrix} 1 & 4 & \tfrac{7}{2} \\ 4 & 8 & 3 \\ \tfrac{7}{2} & 3 & 7 \end{bmatrix}, \qquad \dfrac{1}{2}(A-A^T) = \begin{bmatrix} 0 & -2 & \tfrac{1}{2} \\ 2 & 0 & -2 \\ -\tfrac{1}{2} & 2 & 0 \end{bmatrix}

Adding the two returns $A$ entry by entry.

Key properties

Every diagonal entry of a skew-symmetric matrix is 0, since $a_{ii} = -a_{ii}$ .
A skew-symmetric matrix of odd order has determinant 0. For example, $\det\begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix} = 0$ .
For any square $A$ : $A + A^T$ and $AA^T$ are symmetric, and $A - A^T$ is skew-symmetric.
The only matrix that is both symmetric and skew-symmetric is the zero matrix.

Common mistakes

Forgetting the factor $\tfrac{1}{2}$ in the decomposition, which makes the parts add up to $2A$ .
Calling a matrix skew-symmetric with a nonzero diagonal. Its diagonal must be all zeros.
Transposing incorrectly. Rows become columns: entry $(i, j)$ moves to $(j, i)$ .

Where this comes up in exams

The 2025-26 CBSE Class XII syllabus lists symmetric and skew-symmetric matrices with the transpose, and JEE Advanced 2026 lists symmetric and skew-symmetric matrices and their properties. JEE Main 2026 lists types of matrices without naming these two.

Checked against CBSE Mathematics curriculum, Classes XI-XII (2025-26), JEE (Advanced) 2026 syllabus and JEE (Main) 2026 syllabus (NTA). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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