MathsCBSE Class 12 · Matrices

Symmetric and Skew-Symmetric Matrices (Class 12)

Updated 15 October 2026 · Fact-checked

A square matrix A is symmetric if A′ = A, so entries mirror across the main diagonal, and skew-symmetric if A′ = −A, so mirrored entries are negatives and every diagonal entry is zero. Every square matrix can be written uniquely as ½(A + A′) + ½(A − A′), a symmetric plus a skew-symmetric matrix.

Across the red diagonal, symmetric entries match, while skew-symmetric entries flip sign and the diagonal is all zeros.

Illustration drawn for QuestionBuddy.

Understand Symmetric and Skew-Symmetric Matrices

The transpose $A^{'}$ (also written $A^{T}$ ) of a matrix is obtained by turning rows into columns: the $(i, j)$ entry of $A^{'}$ is the $(j, i)$ entry of $A$ . An $m \times n$ matrix has an $n \times m$ transpose. Four rules are used constantly:

$(A^{'})^{'} = A$
$(A + B)^{'} = A^{'} + B^{'}$
$(k A)^{'} = k A^{'}$ for any scalar $k$
$(A B)^{'} = B^{'} A^{'}$ (the order reverses, like taking off socks and shoes)

Symmetric matrices. A square matrix is symmetric when $A^{'} = A$ , that is $a_{ij} = a_{j i}$ for all $i, j$ . Fold the matrix along its main diagonal and the entries match. The diagonal itself can be anything. Distance tables between towns are symmetric.

Skew-symmetric matrices. A square matrix is skew-symmetric when $A^{'} = - A$ , that is $a_{ij} = - a_{j i}$ . Putting $i = j$ gives $a_{ii} = - a_{ii}$ , so $2 a_{ii} = 0$ : every diagonal entry of a skew-symmetric matrix is zero. Tables of 'net amount owed' or 'goal difference' between pairs are skew-symmetric.

Both at once? If $A^{'} = A$ and $A^{'} = - A$ then $A = - A$ , so $A = O$ . The zero matrix is the only matrix that is both.

The decomposition. For any square matrix $A$ :

$P = \frac{1}{2} (A + A^{'})$ is symmetric, because $P^{'} = \frac{1}{2} (A^{'} + A) = P$ .
$Q = \frac{1}{2} (A - A^{'})$ is skew-symmetric, because $Q^{'} = \frac{1}{2} (A^{'} - A) = - Q$ .
$P + Q = A$ .

This split is unique. Board questions ask you to compute $P$ and $Q$ for a given $3 \times 3$ matrix and verify that they add to $A$ . A quick check: $Q$ must have zeros on the diagonal, and $P$ must have the same diagonal as $A$ .

Useful results with products (prove them with the transpose rules):

$A A^{'}$ and $A^{'} A$ are always symmetric: $(A^{'} A)^{'} = A^{'} (A^{'})^{'} = A^{'} A$ .
If $A$ is symmetric, so is $B^{'} A B$ for any $B$ of suitable order.
If $A$ and $B$ are symmetric, $A B$ is symmetric exactly when $A B = B A$ , and $A B - B A$ is always skew-symmetric.
If $A$ is skew-symmetric, $A^{2}$ is symmetric and $A^{3}$ is skew-symmetric (even powers symmetric, odd powers skew).

Counting. In an $n \times n$ symmetric matrix the free entries are the diagonal and the entries above it: $\frac{n ( n + 1 )}{2}$ of them. A skew-symmetric matrix has only the $\frac{n ( n - 1 )}{2}$ entries above the diagonal free. These counts answer MCQs such as 'how many $3 \times 3$ symmetric matrices have entries 0 or 1?' ( $2^{6} = 64$ ).

This is NCERT Sections 3.5 and 3.6, and a 1-mark question on symmetric or skew-symmetric matrices appears in most board papers.

Symmetric and Skew-Symmetric Matrices: formulas to remember

Transpose of a product: $(A B)^{'} = B^{'} A^{'}$; The order reverses.
Other transpose rules: $(A^{'})^{'} = A, (A + B)^{'} = A^{'} + B^{'}, (k A)^{'} = k A^{'}$; Used in every proof on this topic.
Symmetric: $A^{'} = A ⟺ a_{ij} = a_{j i}$; Mirror image across the main diagonal.
Skew-symmetric: $A^{'} = - A ⟺ a_{ij} = - a_{j i}; a_{ii} = 0$; All diagonal entries are zero.
Decomposition: $A = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})$; Symmetric part plus skew-symmetric part; unique.
Free entries: $symmetric: \frac{n ( n + 1 )}{2}, skew-symmetric: \frac{n ( n - 1 )}{2}$; For $n = 3$ : 6 and 3.

How to solve Symmetric and Skew-Symmetric Matrices questions

Use this for 'express A as the sum of a symmetric and a skew-symmetric matrix', 'find the unknowns', and proof questions.

1Write $A^{'}$ carefully: row 1 of $A$ becomes column 1 of $A^{'}$ , and so on. Check one off-diagonal entry against $A$ .
2Compute $A + A^{'}$ and $A - A^{'}$ entry by entry, then halve each to get $P$ and $Q$ .
3Check: $P$ has the same diagonal as $A$ and mirrors across it; $Q$ has a zero diagonal and opposite mirrored entries.
4Verify $P + Q = A$ and write: ' $P$ is symmetric since $P^{'} = P$ ; $Q$ is skew-symmetric since $Q^{'} = - Q$ .'
5For unknown entries in a symmetric or skew-symmetric matrix, equate each pair $a_{ij}$ and $\pm a_{j i}$ , and set diagonal entries of a skew matrix to zero.
6For proofs, take the transpose of the given expression, apply the four transpose rules (especially $(A B)^{'} = B^{'} A^{'}$ ), and use the given facts $A^{'} = \pm A$ .

Symmetric and Skew-Symmetric Matrices in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Symmetric matrices

Suppose a school bus planner writes the road distances (in km) between three stops X, Y and Z in a table: X to Y is 12 km, X to Z is 20 km and Y to Z is 15 km.

Write the distance matrix D (rows and columns in the order X, Y, Z). Show that D is symmetric and explain why its diagonal is zero.

Try it first, then show the solutionSolution

$D = 012201201520150$ .
Transposing swaps rows and columns: $D^{'} = 012201201520150 = D$ , so D is symmetric.
The distance from Y to X equals the distance from X to Y, which is exactly $d_{ij} = d_{j i}$ .
Each diagonal entry is the distance from a stop to itself, which is 0.

Answer: D is symmetric because road distance does not depend on direction; the diagonal is 0 because each stop is 0 km from itself.

Memory hook: Two-way roads give a symmetric table.

Skew-symmetric matrices

Imagine a three-team football round-robin. Team P beat Q 3–1, Q drew with R 2–2, and R beat P 2–0. In a table, the entry in row $i$ , column $j$ is the goals of team $i$ minus the goals of team $j$ in their match.

Write this goal-difference matrix G, show it is skew-symmetric, and use its row sums to find each team's total goal difference.

Try it first, then show the solutionSolution

Rows and columns in the order P, Q, R: $g_{P Q} = 3 - 1 = 2$ , $g_{QP} = - 2$ ; $g_{QR} = 0 = g_{R Q}$ ; $g_{R P} = 2 - 0 = 2$ , $g_{P R} = - 2$ ; diagonal entries 0.
$G = 0 - 2 2 200 - 2 00$ , and $G^{'} = - G$ since every $g_{ij} = - g_{j i}$ .
Row sums: P: $0 + 2 - 2 = 0$ ; Q: $- 2 + 0 + 0 = - 2$ ; R: $2 + 0 + 0 = 2$ .
All entries add to 0, as they must in a skew-symmetric matrix.

Answer: G is skew-symmetric. Goal differences: P 0, Q $- 2$ , R $+ 2$ .

Memory hook: My goal difference against you is minus yours against me: skew-symmetric.

Tricks to remember Symmetric and Skew-Symmetric Matrices

Short memory aids. Say each one aloud once, then use it on a question the same day.

Fold along the diagonal
Symmetric: folding along the main diagonal lands equal numbers on each other. Skew: folding lands opposite numbers, and the fold line itself is all zeros.
Use it when: Spotting symmetric or skew matrices in MCQs.
Socks and shoes
$(A B)^{'} = B^{'} A^{'}$ : you put on socks then shoes, but take off shoes then socks.
Use it when: Every proof involving the transpose of a product.
Half-sum and half-difference
Symmetric part: average of $A$ and its mirror, $\frac{A + A ^{'}}{2}$ . Skew part: half the difference, $\frac{A - A ^{'}}{2}$ .
Use it when: The decomposition question.
Even powers are friendly
For a skew-symmetric A, even powers ( $A^{2}$ , $A^{4}$ ) are symmetric and odd powers stay skew, just as $(- 1)^{n}$ is $+ 1$ for even n.
Use it when: Assertion-reason items on powers.

Symmetric and Skew-Symmetric Matrices: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: A 2 × 2 decomposition · 2 marks · Very short answerBoard level

Express $A = [2 - 1 74]$ as the sum of a symmetric and a skew-symmetric matrix.

Try it first, then show the solutionSolution

$A^{'} = [27 - 1 4]$ . (½ mark)
$P = \frac{1}{2} (A + A^{'}) = \frac{1}{2} [4668] = [2334]$ , symmetric. (½ mark)
$Q = \frac{1}{2} (A - A^{'}) = \frac{1}{2} [0 - 8 80] = [0 - 4 40]$ , skew-symmetric. (½ mark)
$P + Q = [2 - 1 74] = A$ . (½ mark)

Answer: $A = [2334] + [0 - 4 40]$

Example 2: Unknown entries of a skew-symmetric matrix · 2 marks · Very short answerBoard level

If $A = 02 c a 01 3 b 0$ is skew-symmetric, find $a$ , $b$ and $c$ .

Try it first, then show the solutionSolution

Skew-symmetric means $a_{ij} = - a_{j i}$ . (½ mark)
$a_{12} = - a_{21}$ : $a = - 2$ . (½ mark)
$a_{23} = - a_{32}$ : $b = - 1$ . (½ mark)
$a_{31} = - a_{13}$ : $c = - 3$ . The diagonal is already zero, as required. (½ mark)

Answer: $a = - 2$ , $b = - 1$ , $c = - 3$

Example 3: A 3 × 3 decomposition · 3 marks · Short answerHarder board level

Express $A = 127 45 - 2 - 3 06$ as the sum of a symmetric matrix and a skew-symmetric matrix.

Try it first, then show the solutionSolution

$A^{'} = 14 - 3 250 7 - 2 6$ . (½ mark)
$A + A^{'} = 264 610 - 2 4 - 2 12$ , so $P = 132 35 - 1 2 - 1 6$ , and $P^{'} = P$ . (1 mark)
$A - A^{'} = 0 - 2 10 20 - 2 - 10 20$ , so $Q = 0 - 1 5 10 - 1 - 5 10$ , and $Q^{'} = - Q$ . (1 mark)
Check: $P + Q = 127 45 - 2 - 3 06 = A$ . (½ mark)

Answer: $A = 132 35 - 1 2 - 1 6 + 0 - 1 5 10 - 1 - 5 10$

Example 4: Proofs with the transpose rules · 3 marks · Short answerHarder board level

(i) Prove that $A^{'} A$ is symmetric for every matrix $A$ . (ii) Prove that if $A$ is skew-symmetric, then $A^{2}$ is symmetric and $A^{3}$ is skew-symmetric.

Try it first, then show the solutionSolution

(i) $(A^{'} A)^{'} = A^{'} (A^{'})^{'} = A^{'} A$ , using $(X Y)^{'} = Y^{'} X^{'}$ and $(A^{'})^{'} = A$ . So $A^{'} A$ is symmetric. (1 mark)
(ii) Given $A^{'} = - A$ . Then $(A^{2})^{'} = (AA)^{'} = A^{'} A^{'} = (- A) (- A) = A^{2}$ , so $A^{2}$ is symmetric. (1 mark)
$(A^{3})^{'} = (A^{2} A)^{'} = A^{'} (A^{2})^{'} = (- A) (A^{2}) = - A^{3}$ , so $A^{3}$ is skew-symmetric. (1 mark)

Answer: $(A^{'} A)^{'} = A^{'} A$ ; $(A^{2})^{'} = A^{2}$ ; $(A^{3})^{'} = - A^{3}$ .

Example 5: Counting symmetric matrices · 2 marks · Very short answerHOTS / competency

(i) How many $3 \times 3$ symmetric matrices have every entry equal to 0 or 1? (ii) How many $3 \times 3$ skew-symmetric matrices have every entry from ${- 1, 0, 1}$ ?

Try it first, then show the solutionSolution

(i) A symmetric matrix is fixed by its diagonal (3 entries) and the entries above it (3 entries): 6 free entries, each 0 or 1, giving $2^{6} = 64$ . (1 mark)
(ii) The diagonal must be 0 and the entries below are the negatives of those above, so only the 3 entries above the diagonal are free. Each can be $- 1$ , 0 or 1, and its negative is then also in the set: $3^{3} = 27$ . (1 mark)

Answer: (i) 64 (ii) 27

Example 6: Case study: who owes whom · 4 marks · Case-basedHOTS / competency

Suppose three friends, Aman, Bina and Chirag, settle a trip's expenses. Net of all bills, Aman owes Bina Rs 200, Bina owes Chirag Rs 150, and Chirag owes Aman Rs 50. Let $M$ be the matrix (rows and columns in the order A, B, C) whose $(i, j)$ entry is the net amount $i$ owes $j$ (negative if $j$ owes $i$ ).
(i) Write $M$ . (1 mark)
(ii) Show that $M$ is skew-symmetric. (1 mark)
(iii) Find each friend's overall net amount owed (row sums) and interpret the signs. (2 marks)

Try it first, then show the solutionSolution

(i) $m_{A B} = 200$ , $m_{B A} = - 200$ ; $m_{B C} = 150$ , $m_{C B} = - 150$ ; $m_{C A} = 50$ , $m_{A C} = - 50$ ; diagonal 0. So $M = 0 - 200 50 2000 - 150 - 50 1500$ . (1 mark)
(ii) Every $m_{ij} = - m_{j i}$ and the diagonal is 0, so $M^{'} = - M$ . (1 mark)
(iii) Row sums: Aman $200 - 50 = 150$ ; Bina $- 200 + 150 = - 50$ ; Chirag $50 - 150 = - 100$ . (1 mark)
Aman must pay Rs 150 in total; Bina receives Rs 50 and Chirag receives Rs 100. The sums add to 0. (1 mark)

Answer: (i) $M = 0 - 200 50 2000 - 150 - 50 1500$ . (ii) $M^{'} = - M$ . (iii) Aman owes Rs 150 net; Bina is owed Rs 50; Chirag is owed Rs 100.

Where students lose marks in Symmetric and Skew-Symmetric Matrices

The slips examiners see most often, and what to write instead.

Mistake: Writing (AB)′ = A′B′
The transpose of a product reverses the order.
Wrong working
$(A^{'} A)^{'} = (A^{'})^{'} A^{'} = A A^{'}$ , so not symmetric.
Right working
$(A^{'} A)^{'} = A^{'} (A^{'})^{'} = A^{'} A$ , so it is symmetric.
Do this: Always reverse the order when transposing a product.
Usually costs: The whole proof (2 to 3 marks).
Mistake: Forgetting to halve
$A + A^{'}$ is symmetric, but $(A + A^{'}) + (A - A^{'}) = 2 A$ , not $A$ .
Wrong working
$A = (A + A^{'}) + (A - A^{'})$ .
Right working
$A = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})$ .
Do this: After computing, add $P$ and $Q$ and check that you get $A$ .
Usually costs: 1 to 2 marks.
Mistake: Transposing incorrectly
Students swap only some entries, or reflect in the wrong diagonal.
Wrong working
$127 45 - 2 - 3 06^{'} = 60 - 3 - 2 54 721$ .
Right working
Row 1 becomes column 1: $A^{'} = 14 - 3 250 7 - 2 6$ .
Do this: Copy each row of A down a column of A′, in order.
Usually costs: Everything after the transpose.
Mistake: Not stating why P and Q have the required type
The question says 'express as the sum of a symmetric and a skew-symmetric matrix'; the claim must be justified.
Wrong working
$P = \dots$ , $Q = \dots$ , $P + Q = A$ .
Right working
Add: ' $P^{'} = P$ , so P is symmetric; $Q^{'} = - Q$ , so Q is skew-symmetric.'
Do this: Write one line for each property.
Usually costs: Half to one mark.

Exam tips

1-mark questions on skew-symmetric matrices appeared in the 2024 and 2026 board papers and the 2025 and 2026 sample papers; remember the zero diagonal and $a_{ij} = - a_{j i}$ .
In the decomposition question, present $A^{'}$ , $P$ and $Q$ as separate displayed matrices with the factor $\frac{1}{2}$ shown.
Fractions in P and Q are fine; do not multiply them away.
For proofs, write the transpose rule you use on each line; examiners award marks for the reasoning, not just the result.
If a question says 'A is both symmetric and skew-symmetric', the answer is immediately $A = O$ .

Practice questions on Symmetric and Skew-Symmetric Matrices

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If A is a skew-symmetric matrix, then each diagonal entry of A is:
(a) 1
(b) 0
(c) $- 1$
(d) any real number
Hint
Put $i = j$ in $a_{ij} = - a_{j i}$ .
Answer
(b) 0, since $a_{ii} = - a_{ii}$ gives $2 a_{ii} = 0$ .
Q2 · 1 mark · MCQBoard level
If $[23 x 5]$ is symmetric, then $x$ equals:
(a) 2
(b) 3
(c) 5
(d) $- 3$
Hint
Symmetric means $a_{12} = a_{21}$ .
Answer
(b) 3.
Q3 · 2 marks · Very short answerBoard level
For $A = [47 - 3 2]$ , show that $A + A^{'}$ is symmetric and $A - A^{'}$ is skew-symmetric.
Hint
Compute both matrices, then take their transposes.
Answer
$A + A^{'} = [8444]$ , which equals its transpose, so it is symmetric. $A - A^{'} = [010 - 10 0]$ , whose transpose is $[0 - 10 100] = - (A - A^{'})$ , so it is skew-symmetric.
Q4 · 3 marks · Short answerHarder board level
Express $A = 24 - 1 016 3 - 2 5$ as the sum of a symmetric and a skew-symmetric matrix.
Hint
Check that your skew part has a zero diagonal.
Answer
$A^{'} = 203 41 - 2 - 1 65$ . $P = \frac{1}{2} (A + A^{'}) = 221212125$ , $Q = \frac{1}{2} (A - A^{'}) = 02 - 2 - 2 04 2 - 4 0$ , and $P + Q = A$ .
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The matrix $[0 - 2 20]$ is both symmetric and skew-symmetric.
Reason (R): The only matrix that is both symmetric and skew-symmetric is a zero matrix.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Compare the (1, 2) and (2, 1) entries.
Answer
(d). The matrix is skew-symmetric but not symmetric ( $2 \neq = - 2$ ), so A is false. R is true: $A^{'} = A$ and $A^{'} = - A$ give $A = O$ .
Q6 · 3 marks · Short answerHarder board level
If A and B are symmetric matrices of the same order, prove that AB is symmetric if and only if $A B = B A$ .
Hint
Start from $(A B)^{'} = B^{'} A^{'}$ and use $A^{'} = A$ , $B^{'} = B$ .
Answer
$(A B)^{'} = B^{'} A^{'} = B A$ . If $A B = B A$ , then $(A B)^{'} = A B$ , so AB is symmetric. Conversely, if AB is symmetric, then $A B = (A B)^{'} = B A$ . Hence AB is symmetric exactly when $A B = B A$ .
Q7 · 3 marks · Short answerHOTS / competency
(i) How many $3 \times 3$ skew-symmetric matrices have entries from ${- 1, 0, 1}$ ? (ii) How many of these are also symmetric? (iii) How many $2 \times 2$ symmetric matrices have entries from ${0, 1}$ ?
Hint
Count only the free entries in each case.
Answer
(i) 3 free entries above the diagonal, each with 3 choices: $3^{3} = 27$ . (ii) Only the zero matrix is both, so 1. (iii) A $2 \times 2$ symmetric matrix has 3 free entries ( $a_{11}, a_{12}, a_{22}$ ): $2^{3} = 8$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is $(A B)^{'}$ ?Show answer
$B^{'} A^{'}$ .
2.When is a square matrix skew-symmetric?Show answer
When $A^{'} = - A$ , that is $a_{ij} = - a_{j i}$ for all $i, j$ .
3.Write the symmetric part of a square matrix A.Show answer
$\frac{1}{2} (A + A^{'})$ .
4.If A is skew-symmetric, what kind of matrix is $A^{2}$ ?Show answer
Symmetric.
5.Which matrix is both symmetric and skew-symmetric?Show answer
Only the zero matrix.
6.How many free entries does a $4 \times 4$ symmetric matrix have?Show answer
$\frac{4 \times 5}{2} = 10$ .

Symmetric and Skew-Symmetric Matrices: frequently asked questions

Why are the diagonal entries of a skew-symmetric matrix zero?

Skew-symmetry needs each entry to be the negative of its mirror image. A diagonal entry is its own mirror image, so it must equal its own negative, and the only such number is zero.

Can every square matrix be written as a symmetric plus a skew-symmetric matrix?

Yes. Take P = ½(A + A′) and Q = ½(A − A′). Then P is symmetric, Q is skew-symmetric and P + Q = A. The split is unique.

Is a diagonal matrix symmetric?

Yes. All its off-diagonal entries are zero, so each one equals its mirror image. A diagonal matrix is skew-symmetric only if it is the zero matrix.

Which NCERT sections cover this topic?

Section 3.5 (Transpose of a Matrix) and Section 3.6 (Symmetric and Skew Symmetric Matrices) of Chapter 3, Matrices, in NCERT Mathematics Part I for Class 12.

Can a non-square matrix be symmetric?

No. A symmetric or skew-symmetric matrix must equal plus or minus its own transpose, which has the same order only when the matrix is square.