MathsISC Class 12 · Matrices

Symmetric and Skew-Symmetric Matrices: Properties, Proofs and Decomposition

Updated 15 October 2026 · Fact-checked

A square matrix A is symmetric if its transpose equals itself (A′ = A), so entries mirror across the leading diagonal. It is skew-symmetric if A′ = −A, which forces every diagonal entry to be zero. Every square matrix splits uniquely as ½(A + A′) plus ½(A − A′): a symmetric part plus a skew-symmetric part.

Understand Symmetric and Skew-Symmetric Matrices

The transpose $A^{'}$ (also written $A^{T}$ ) of a matrix is formed 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 do almost all the work in this topic:

$(A^{'})^{'} = A$
$(A + B)^{'} = A^{'} + B^{'}$
$(k A)^{'} = k A^{'}$ for any scalar $k$
$(A B)^{'} = B^{'} A^{'}$ , with the order reversed.

The reversal in the last rule is the one students forget. It follows from the fact that the $(i, j)$ entry of $(A B)^{'}$ is row $j$ of $A$ times column $i$ of $B$ , which is exactly row $i$ of $B^{'}$ times column $j$ of $A^{'}$ .

Symmetric matrices. A square matrix is symmetric when $A^{'} = A$ , that is $a_{ij} = a_{j i}$ for all $i, j$ . Its entries are mirror images across the leading diagonal, and the diagonal itself can be anything. A distance table between towns is a natural example.

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 is zero. The entries above the diagonal are the negatives of those below. To find an unknown entry that makes a matrix skew-symmetric, simply match each entry with the negative of its mirror image. The ISC 2025 examiners noted that candidates confused symmetric with skew-symmetric in exactly this kind of question.

Building symmetric and skew-symmetric matrices. For any square matrix $A$ :

$A + A^{'}$ is symmetric, because $(A + A^{'})^{'} = A^{'} + A$ .
$A - A^{'}$ is skew-symmetric, because $(A - A^{'})^{'} = A^{'} - A = - (A - A^{'})$ .
$A A^{'}$ and $A^{'} A$ are symmetric (for any matrix, square or not), because $(A^{'} A)^{'} = A^{'} (A^{'})^{'} = A^{'} A$ .

Decomposition. Since $A = \frac{1}{2} (A + A^{'}) + \frac{1}{2} (A - A^{'})$ , every square matrix is the sum of a symmetric matrix $P = \frac{1}{2} (A + A^{'})$ and a skew-symmetric matrix $Q = \frac{1}{2} (A - A^{'})$ . The split is unique: if $A = P_{1} + Q_{1}$ as well, then $P - P_{1} = Q_{1} - Q$ is both symmetric and skew-symmetric, so it equals its own negative and is the zero matrix.

Products. If $A$ and $B$ are symmetric of the same order, then $(A B)^{'} = B^{'} A^{'} = B A$ . So $A B$ is symmetric exactly when $A B = B A$ ; $A B + B A$ is always symmetric, and $A B - B A$ is always skew-symmetric (the ISC 2024 MCQ that the examiners said most candidates got wrong). If $A$ is skew-symmetric, $A^{2}$ is symmetric, since $(A^{2})^{'} = (A^{'})^{2} = (- A)^{2} = A^{2}$ .

Determinant link. If $A$ is a skew-symmetric matrix of odd order, $∣ A ∣ = ∣ A^{'} ∣ = ∣ - A ∣ = (- 1)^{n} ∣ A ∣ = - ∣ A ∣$ , so $∣ A ∣ = 0$ . For example, every $3 \times 3$ skew-symmetric matrix is singular.

Symmetric and Skew-Symmetric Matrices: formulas to remember

Transpose rules: $(A^{'})^{'} = A, (A + B)^{'} = A^{'} + B^{'}, (k A)^{'} = k A^{'}$; Apply entry by entry.
Reversal law: $(A B)^{'} = B^{'} A^{'}$; The order reverses; for three matrices (ABC)′ = C′B′A′.
Symmetric: $A^{'} = A ⟺ a_{ij} = a_{j i}$; Mirror image across the leading 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; the split is unique.
Products of symmetric matrices: $A, B symmetric \Rightarrow A B + B A symmetric, A B - B A skew-symmetric$; AB itself is symmetric if and only if AB = BA.
Odd-order skew-symmetric: $A^{'} = - A, n odd \Rightarrow ∣ A ∣ = 0$; From |A′| = |A| and |−A| = (−1)ⁿ|A|.

How to solve Symmetric and Skew-Symmetric Matrices questions

ISC questions on this topic are of three kinds: find unknown entries, express a matrix as P + Q, and prove a property using transpose rules. Here is the method for each.

1Unknown entries: write the conditions $a_{ij} = a_{j i}$ (symmetric) or $a_{ij} = - a_{j i}$ and $a_{ii} = 0$ (skew-symmetric) for each pair containing an unknown, and solve.
2Decomposition: write $A$ and $A^{'}$ side by side, then compute $A + A^{'}$ and $A - A^{'}$ entry by entry.
3Halve each to get $P = \frac{1}{2} (A + A^{'})$ and $Q = \frac{1}{2} (A - A^{'})$ . Check that $P$ mirrors across the diagonal and $Q$ has zeros on the diagonal.
4Add $P$ and $Q$ to confirm you get $A$ back; state "P is symmetric, Q is skew-symmetric and A = P + Q".
5Proofs: take the transpose of the given expression, apply $(A + B)^{'} = A^{'} + B^{'}$ and $(A B)^{'} = B^{'} A^{'}$ , then replace $A^{'}$ by $A$ or $- A$ as given.
6Compare the result with the original expression: if you get the same expression it is symmetric; if you get its negative it is skew-symmetric.
7For MCQs on products, test with the transpose rule rather than with numbers; it takes one line.

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 matrix

Suppose a delivery firm stores the road distances (in km) between its three depots P, Q and R in a matrix $D$ , where $d_{ij}$ is the distance from depot $i$ to depot $j$ : $D = 012151209 x 90$ .

Explain why $D$ should be symmetric, find $x$ , and say why the diagonal entries are zero.

Try it first, then show the solutionSolution

The road distance from P to R is the same as from R to P, so $d_{ij} = d_{j i}$ for every pair: $D$ must equal $D^{'}$ .
Comparing $d_{13} = x$ with $d_{31} = 15$ gives $x = 15$ .
Each diagonal entry is the distance from a depot to itself, which is 0. (A symmetric matrix may have non-zero diagonal entries in general; here they are zero because of what they measure.)

Answer: $x = 15$ ; $D$ is symmetric because distances are the same both ways, and its diagonal is zero because each depot is 0 km from itself.

Memory hook: A distance table is a mirror: the same both ways across the diagonal.

Skew-symmetric matrix of net balances

Imagine three flatmates. In a month, $m_{ij}$ is the amount (in rupees) that flatmate $i$ paid on behalf of flatmate $j$ : $M = 01005030002501502000$ .

Find $N = M - M^{'}$ , show that it is skew-symmetric, and use it to find each flatmate's net position.

Try it first, then show the solutionSolution

$M^{'} = 03001501000200502500$ , so $N = M - M^{'} = 0 - 200 - 100 200050 100 - 50 0$ .
$N^{'} = - N$ and the diagonal is zero, so $N$ is skew-symmetric. Entry $n_{12} = 200$ means flatmate 2 owes flatmate 1 a net Rs 200, and $n_{21} = - 200$ says the same from the other side.
Row sums give net positions: flatmate 1: $200 + 100 = 300$ ; flatmate 2: $- 200 - 50 = - 250$ ; flatmate 3: $- 100 + 50 = - 50$ .
The row sums add to zero, as they must: money is only moved between the flatmates.

Answer: Flatmate 1 is owed Rs 300 in all; flatmate 2 owes Rs 250 and flatmate 3 owes Rs 50.

Memory hook: What I owe you is minus what you owe me: a net-balance table is 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.

Mirror versus anti-mirror
Symmetric: fold along the diagonal and the entries match. Skew-symmetric: fold and they match with opposite signs, and the crease (diagonal) is all zeros.
Use it when: Finding unknown entries quickly.
Socks and shoes again
(AB)′ = B′A′: take off in reverse order.
Use it when: Every proof that involves a product.
Plus is symmetric, minus is skew
A + A′ and AB + BA (A, B symmetric) are symmetric; A − A′ and AB − BA are skew-symmetric.
Use it when: One-mark MCQs on properties.
Odd skew is singular
An odd-order skew-symmetric matrix always has determinant 0.
Use it when: Spotting determinant values without expanding.

Symmetric and Skew-Symmetric Matrices: solved examples

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

Example 1: Unknown entries of a skew-symmetric matrix · 1 mark · Very short answerBoard level

Find $a$ and $b$ if $A = 02 - 3 a 0 - 1 3 b 0$ is skew-symmetric.

Try it first, then show the solutionSolution

Skew-symmetric needs $a_{12} = - a_{21}$ : $a = - 2$ .
And $a_{23} = - a_{32}$ : $b = 1$ . (Also $a_{13} = 3 = - a_{31}$ and the diagonal is zero, as required.)

Answer: $a = - 2$ , $b = 1$ .

Example 2: Splitting a 3 × 3 matrix · 3 marks · Short answerBoard level

Express $A = 12 - 3 431 - 1 50$ as the sum of a symmetric and a skew-symmetric matrix.

Try it first, then show the solutionSolution

$A^{'} = 14 - 1 235 - 3 10$ . (½ mark)
$P = \frac{1}{2} (A + A^{'}) = \frac{1}{2} 26 - 4 666 - 4 60 = 13 - 2 333 - 2 30$ , and $P^{'} = P$ . (1 mark)
$Q = \frac{1}{2} (A - A^{'}) = \frac{1}{2} 0 - 2 - 2 20 - 4 240 = 0 - 1 - 1 10 - 2 120$ , and $Q^{'} = - Q$ . (1 mark)
Check: $P + Q = 12 - 3 431 - 1 50 = A$ . (½ mark)

Answer: $A = 13 - 2 333 - 2 30 + 0 - 1 - 1 10 - 2 120$ .

Example 3: Products of symmetric matrices · 1 mark · MCQHarder board level

If $A$ and $B$ are symmetric matrices of the same order, then $A B + B A$ is:
(a) skew-symmetric
(b) symmetric
(c) a zero matrix
(d) an identity matrix

Try it first, then show the solutionSolution

$(A B + B A)^{'} = B^{'} A^{'} + A^{'} B^{'} = B A + A B$ , which is the same matrix.

Answer: (b) symmetric. (By the same working, $A B - B A$ is skew-symmetric, the fact tested in ISC 2024.)

Example 4: A'A and the square of a skew-symmetric matrix · 2 marks · Short answerHarder board level

(i) Show that $A^{'} A$ is symmetric for every matrix $A$ . (ii) If $K$ is skew-symmetric, show that $K^{2}$ is symmetric.

Try it first, then show the solutionSolution

(i) $(A^{'} A)^{'} = A^{'} (A^{'})^{'} = A^{'} A$ , so $A^{'} A$ is symmetric. (1 mark)
(ii) $(K^{2})^{'} = (K K)^{'} = K^{'} K^{'} = (- K) (- K) = K^{2}$ , so $K^{2}$ is symmetric. (1 mark)

Answer: Both proved using the reversal law.

Example 5: Mixing symmetric and skew-symmetric · 3 marks · Short answerHarder board level

If $A$ is symmetric and $B$ is skew-symmetric, both of order $n$ , show that $A B - B A$ is symmetric and $A B + B A$ is skew-symmetric.

Try it first, then show the solutionSolution

Given $A^{'} = A$ and $B^{'} = - B$ . (½ mark)
$(A B - B A)^{'} = B^{'} A^{'} - A^{'} B^{'} = (- B) A - A (- B) = - B A + A B = A B - B A$ . So $A B - B A$ is symmetric. (1½ marks)
$(A B + B A)^{'} = B^{'} A^{'} + A^{'} B^{'} = - B A - A B = - (A B + B A)$ . So $A B + B A$ is skew-symmetric. (1 mark)

Answer: Proved.

Example 6: Three facts about skew-symmetric matrices · 5 marks · Long answerHOTS / competency

(i) Prove that every diagonal entry of a skew-symmetric matrix is zero.
(ii) Prove that a $3 \times 3$ skew-symmetric matrix has determinant 0.
(iii) Prove that the expression of a square matrix as the sum of a symmetric and a skew-symmetric matrix is unique.

Try it first, then show the solutionSolution

(i) $A^{'} = - A$ gives $a_{ij} = - a_{j i}$ for all $i, j$ . With $i = j$ : $a_{ii} = - a_{ii}$ , so $2 a_{ii} = 0$ and $a_{ii} = 0$ . (1 mark)
(ii) $∣ A ∣ = ∣ A^{'} ∣ = ∣ - A ∣$ . Multiplying each of the 3 rows by $- 1$ multiplies the determinant by $(- 1)^{3}$ , so $∣ - A ∣ = - ∣ A ∣$ . Hence $∣ A ∣ = - ∣ A ∣$ and $∣ A ∣ = 0$ . (2 marks)
(iii) Suppose $A = P + Q = P_{1} + Q_{1}$ with $P, P_{1}$ symmetric and $Q, Q_{1}$ skew-symmetric. Then $S = P - P_{1} = Q_{1} - Q$ . (1 mark)
$S$ is symmetric (difference of symmetric matrices) and skew-symmetric (difference of skew-symmetric matrices), so $S = S^{'} = - S$ , giving $S = O$ . Hence $P = P_{1}$ and $Q = Q_{1}$ . (1 mark)

Answer: All three proved; the unique split is $P = \frac{1}{2} (A + A^{'})$ , $Q = \frac{1}{2} (A - A^{'})$ .

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; matrix multiplication is not commutative, so A′B′ is usually different.
Wrong working
$(A B - B A)^{'} = A^{'} B^{'} - B^{'} A^{'} = A B - B A$ , so it is symmetric.
Right working
$(A B - B A)^{'} = B^{'} A^{'} - A^{'} B^{'} = B A - A B = - (A B - B A)$ , so it is skew-symmetric.
Do this: Write the reversal law at the start of every transpose proof.
Usually costs: The whole proof, or the MCQ that most candidates missed in ISC 2024.
Mistake: Confusing symmetric and skew-symmetric
In unknown-entry questions, using a_ij = a_ji for a skew-symmetric matrix gives the wrong sign.
Do this: Skew means a_ij = −a_ji with zeros on the diagonal; check the diagonal first.
Usually costs: 1 mark (reported by the ISC 2025 examiners).
Mistake: Forgetting the ½
A + A′ and A − A′ add up to 2A, not A.
Do this: Halve both parts and check that P + Q = A.
Usually costs: 1 mark.
Mistake: Transposing by swapping the wrong entries
Students sometimes reflect across the other diagonal or swap only some entries.
Do this: Write the first row of A as the first column of A′, then the second row as the second column, and so on.
Usually costs: Every later step in the question.

Exam tips

In a decomposition answer, show $A^{'}$ , then $P$ and $Q$ , then the check $P + Q = A$ . Each line carries credit.
State "since $P^{'} = P$ , P is symmetric" and "since $Q^{'} = - Q$ , Q is skew-symmetric"; do not leave the properties unstated.
For property MCQs, a one-line transpose argument is faster and safer than testing with numbers.
Remember that a symmetric matrix can have any diagonal entries; only skew-symmetric matrices need zeros there.
When a question involves a skew-symmetric matrix of odd order and a determinant, the value 0 is usually the key.

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 any square matrix, then $A - A^{'}$ is:
(a) symmetric
(b) skew-symmetric
(c) a diagonal matrix
(d) the zero matrix
Hint
Take its transpose.
Answer
(b) skew-symmetric, since $(A - A^{'})^{'} = A^{'} - A$ .
Q2 · 1 mark · Very short answerBoard level
Find $x$ if $[4 x + 2 2 x - 1 7]$ is symmetric.
Hint
Set the off-diagonal entries equal.
Answer
$2 x - 1 = x + 2 \Rightarrow x = 3$ .
Q3 · 2 marks · Short answerHarder board level
For $A = [13 2 - 1]$ and $B = [0214]$ , verify that $(A B)^{'} = B^{'} A^{'}$ .
Hint
Compute AB first.
Answer
$A B = [4 - 2 9 - 1]$ , so $(A B)^{'} = [49 - 2 - 1]$ . $B^{'} A^{'} = [0124] [12 3 - 1] = [49 - 2 - 1]$ . Equal.
Q4 · 3 marks · Short answerHarder board level
Express $A = 352 - 1 26 04 - 1$ as $P + Q$ with $P$ symmetric and $Q$ skew-symmetric.
Hint
Compute $\frac{1}{2} (A + A^{'})$ and $\frac{1}{2} (A - A^{'})$ .
Answer
$P = 321225 15 - 1$ , $Q = 031 - 3 01 - 1 - 1 0$ ; $P + Q = A$ .
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): For any square matrix $A$ , the matrix $A + A^{'}$ is symmetric.
Reason (R): $(A^{'})^{'} = A$ and $(A + B)^{'} = A^{'} + B^{'}$ .
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
Hint
Use R to transpose A + A′.
Answer
(a). $(A + A^{'})^{'} = A^{'} + (A^{'})^{'} = A^{'} + A$ , using exactly the two rules in R.
Q6 · 3 marks · Short answerHOTS / competency
If $A$ and $B$ are symmetric matrices of the same order, prove that $A B$ is symmetric if and only if $A B = B A$ .
Hint
Compute $(A B)^{'}$ and compare.
Answer
$(A B)^{'} = B^{'} A^{'} = B A$ . If $A B$ is symmetric, $A B = (A B)^{'} = B A$ . Conversely, if $A B = B A$ , then $(A B)^{'} = B A = A B$ , so $A B$ is symmetric.
Q7 · 4 marks · Case-basedHOTS / competency
Case study: Suppose an airline network links three cities, and $F = 051304240$ gives the number of daily flights from city $i$ to city $j$ .
(i) Is $F$ symmetric? What would symmetry mean here? (1 mark)
(ii) Find the symmetric part $S = \frac{1}{2} (F + F^{'})$ . (1 mark)
(iii) Find the skew-symmetric part $K = \frac{1}{2} (F - F^{'})$ . (1 mark)
(iv) Interpret the entry $k_{12}$ . (1 mark)
Hint
S averages the two directions; K measures the imbalance.
Answer
(i) No ( $f_{12} = 3$ , $f_{21} = 5$ ). Symmetry would mean the same number of flights each way on every route. (ii) $S = 04 1.5 404 1.5 40$ . (iii) $K = 01 - 0.5 - 1 00 0.5 00$ . (iv) $k_{12} = - 1$ : city 1 to city 2 has 1 flight fewer than the two-way average of 4, while city 2 to city 1 has 1 more.

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.State the reversal law for the transpose of a product.Show answer
$(A B)^{'} = B^{'} A^{'}$ .
2.What are the diagonal entries of a skew-symmetric matrix?Show answer
All zero.
3.Write the symmetric part of a square matrix $A$ .Show answer
$\frac{1}{2} (A + A^{'})$ .
4.If $A$ and $B$ are symmetric, what kind of matrix is $A B - B A$ ?Show answer
Skew-symmetric.
5.Can a matrix be both symmetric and skew-symmetric?Show answer
Only the zero matrix, since $A = A^{'} = - A$ gives $A = O$ .
6.What is the determinant of a $3 \times 3$ skew-symmetric matrix?Show answer
0.

Symmetric and Skew-Symmetric Matrices: frequently asked questions

What is the difference between a symmetric and a skew-symmetric matrix?

A symmetric matrix equals its transpose, so entries mirror across the diagonal. A skew-symmetric matrix equals the negative of its transpose, so mirror entries have opposite signs and every diagonal entry is zero.

Is every diagonal matrix symmetric?

Yes. All its off-diagonal entries are zero, so a_ij = a_ji = 0 for i ≠ j. A diagonal matrix is skew-symmetric only if it is the zero matrix.

Why can every square matrix be split into symmetric and skew-symmetric parts?

Because A = ½(A + A′) + ½(A − A′), and the first part is always symmetric while the second is always skew-symmetric. The split is also unique, so there is only one correct answer.

Is the product of two symmetric matrices symmetric?

Not always. If A and B are symmetric, (AB)′ = BA, so AB is symmetric only when AB = BA. AB + BA is always symmetric and AB − BA always skew-symmetric.

Are symmetric and skew-symmetric matrices in the ISC 2027 syllabus?

Yes. The CISCE 2027 syllabus lists the transpose, symmetric and skew-symmetric matrices and their properties under Matrices, and both 2024 and 2025 papers asked about them in Section A.