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Symmetric and skew-symmetric matrices take up only a few pages of the Class 12 matrices chapter, but they appear every year in board papers and turn up as quick MCQs in entrance tests. The ideas are simple; the marks are lost in arithmetic and in a few results students half-remember. This post covers the definitions, a full worked example, the results on products and determinants, and the common slips.
Where is this topic in the syllabus?
The CBSE Mathematics curriculum for 2026-27 lists "transpose of a matrix, symmetric and skew symmetric matrices" in the Class 12 matrices unit. The NCERT Class 12 Matrices chapter covers them in Section 3.6. For entrance exams, the JEE (Main) 2026 syllabus mentions only "type of matrices" in its matrices unit, while the JEE (Advanced) 2026 syllabus explicitly lists "diagonal, symmetric and skew-symmetric matrices and their properties". So the topic is directly tested in boards and Advanced, and fair game in Main under matrix types.
What are symmetric and skew-symmetric matrices?
NCERT's definitions, for a square matrix A = [aᵢⱼ] with transpose A′:
Symmetric: A′ = A, that is aᵢⱼ = aⱼᵢ for all i and j. The matrix is a mirror image across its main diagonal.
Skew-symmetric: A′ = −A, that is aⱼᵢ = −aᵢⱼ. Putting i = j gives aᵢᵢ = −aᵢᵢ, so 2aᵢᵢ = 0 and every diagonal entry is zero.
Only square matrices can be either. The zero matrix is both.
What do the two NCERT theorems say?
Theorem 1. For any square matrix A, A + A′ is symmetric and A − A′ is skew-symmetric. The proof uses the transpose rules (A′)′ = A and (A + B)′ = A′ + B′: (A + A′)′ = A′ + A, and (A − A′)′ = A′ − A = −(A − A′).
Theorem 2. Every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix:
A = ½(A + A′) + ½(A − A′).
The split is also unique. If A = P + Q with P symmetric and Q skew-symmetric, then A′ = P − Q. Adding and subtracting the two equations gives P = ½(A + A′) and Q = ½(A − A′), so there is no other choice.
Worked example: splitting a 3 × 3 matrix
Take A with rows (2, 4, −1), (0, 3, 5), (7, −2, 1).
Step 1: write the transpose. A′ has rows (2, 0, 7), (4, 3, −2), (−1, 5, 1). Each row of A′ is a column of A.
Step 2: the symmetric part P = ½(A + A′).
A + A′ has rows (4, 4, 6), (4, 6, 3), (6, 3, 2).
P has rows (2, 2, 3), (2, 3, 3/2), (3, 3/2, 1).
Check: P is a mirror image across the diagonal, so P′ = P.
Step 3: the skew-symmetric part Q = ½(A − A′).
A − A′ has rows (0, 4, −8), (−4, 0, 7), (8, −7, 0).
Q has rows (0, 2, −4), (−2, 0, 7/2), (4, −7/2, 0).
Check: the diagonal is zero, and each entry is the negative of its mirror entry.
Step 4: verify. P + Q has rows (2, 4, −1), (0, 3, 5), (7, −2, 1), which is A.
What happens with products of symmetric matrices?
Using (AB)′ = B′A′:
If A and B are symmetric, (AB)′ = B′A′ = BA. So AB is symmetric only when AB = BA. NCERT proves this as Example 24.
AB − BA is always skew-symmetric (NCERT Exercise 3.3, question 11). Example: A with rows (1, 2), (2, 3) and B with rows (0, 1), (1, 4) give AB with rows (2, 9), (3, 14) and BA with rows (2, 3), (9, 14). So AB is not symmetric, and AB − BA has rows (0, 6), (−6, 0), which is skew-symmetric.
For any matrix A, A′A is symmetric, since (A′A)′ = A′(A′)′ = A′A.
Why is the determinant of an odd-order skew-symmetric matrix zero?
This result appears often in JEE-style questions. If Q is skew-symmetric of order n, then Q′ = −Q. Taking determinants, and using |Q′| = |Q| and |−Q| = (−1)ⁿ|Q|, gives |Q| = (−1)ⁿ|Q|. For odd n this means |Q| = −|Q|, so |Q| = 0.
The Q from the worked example is 3 × 3, so its determinant must be 0. Expanding along the first row: 0 × (0 × 0 − (7/2)(−7/2)) − 2 × ((−2)(0) − (7/2)(4)) + (−4) × ((−2)(−7/2) − 0 × 4) = 0 − 2 × (−14) − 4 × 7 = 28 − 28 = 0. It checks.
Which mistakes cost marks?
Forgetting the ½. A + A′ is symmetric, but it is 2P, not P.
Non-zero diagonal in a skew part. If you get one, you have made an arithmetic slip.
Writing (AB)′ = A′B′. The order reverses: (AB)′ = B′A′.
Assuming the product of symmetric matrices is symmetric. Only if they commute.
Applying the definitions to non-square matrices. They only apply to square ones.
For the determinant properties that these questions often combine with, see our post on matrices and determinants for JEE.
A practice routine
Write any 3 × 3 matrix with mixed signs and split it, using the two quick checks.
Do NCERT Exercise 3.3 questions 7 to 11 without looking at the text.
Prove three results in your own words: the uniqueness of the split, AB − BA is skew-symmetric, and the odd-order determinant result.
Mix two matrix-type MCQs into each maths set for a fortnight.
If a proof step does not click, Fermi's AI tutor writes on a board and checks understanding as it goes, so you can work through the transpose rules one line at a time.
Pick a 3 × 3 matrix now, split it into symmetric and skew-symmetric parts, and verify that they add back to the original.
