Definition formula properties and solved examples of symmetric and skew symmetric matrices
The concept of symmetric and skew symmetric matrix is a fundamental part of linear algebra and matrices, helping students solve a wide range of mathematical problems and build a solid foundation for advanced exam topics like CBSE, JEE, NEET, and more. Knowing how to identify and use these matrices can help boost accuracy and speed in various competitive exams.
What Is Symmetric Matrix And Skew Symmetric Matrix?
A symmetric matrix is a square matrix in which the element at position (i, j) is the same as the element at (j, i) for all valid i and j. Mathematically, for a matrix A to be symmetric, it must satisfy the condition A = Aᵗ, where Aᵗ is the transpose of A.
A skew symmetric matrix (sometimes called anti-symmetric matrix) is also a square matrix, but in this case, each element at (i, j) is the negative of the element at (j, i). In other words, A = -Aᵗ for skew symmetric matrices.
These matrices are widely used in subjects like Physics, Computer Science (especially in algorithms and data structures), and have direct exam relevance in topics like vector spaces, quadratic forms, and matrix transformations.
Key Formula for Symmetric and Skew Symmetric Matrix
Here are the key formulas for quick reference:
- Symmetric Matrix: A = Aᵗ
- Skew Symmetric Matrix: A = -Aᵗ
In both cases, the matrix A must be a square matrix (same number of rows and columns).
Properties and Key Differences
| Property | Symmetric Matrix | Skew Symmetric Matrix |
|---|---|---|
| Definition | A = Aᵗ | A = -Aᵗ |
| Diagonal Elements | Can be any real/complex number | Always zero (for real matrices) |
| Main Usage | Physics (inertia tensor), quadratic forms, eigenvalues are real | Vectors, rotation matrices, useful in cross products and determinants |
| Example | [[2, 3], [3, 4]] | [[0, 5], [-5, 0]] |
- Both require square shape (n × n).
- The transpose of a symmetric matrix is itself; for skew symmetric, it’s the negative.
- Every square matrix can be written as a sum of a symmetric and a skew symmetric matrix!
Examples with Stepwise Checking
Example 1 (Symmetric Matrix, 2x2):
| Given Matrix A | = | [[4, 7], [7, 5]] |
Stepwise Check:
1. Find Aᵗ (Transpose): Swap rows and columns.2. Aᵗ = [[4, 7], [7, 5]]
3. Since A = Aᵗ, A is symmetric.
Example 2 (Skew Symmetric Matrix, 3x3):
| Given Matrix B | = |
[[0, 3, -4], [-3, 0, 5], [4, -5, 0]] |
Stepwise Check:
1. Transpose B: swap rows/cols2. Bᵗ = [[0, -3, 4], [3, 0, -5], [-4, 5, 0]]
3. Now, check Bᵗ = -B:
- The (1,2) entry in Bᵗ is -3, in B it is 3; similarly for other entries.
4. Yes: Bᵗ = -B ⇒ B is skew symmetric.
Sum of Symmetric and Skew Symmetric Matrix
Any square matrix M can be decomposed as:
M = S + K, where:
K = (1/2)(M – Mᵗ) (Skew symmetric part)
Let’s see this with an example.
Given: M = [[2, 4], [7, 6]]
1. Find Mᵗ = [[2, 7], [4, 6]]2. Symmetric part S = ½(M + Mᵗ) = ½([[2+2, 4+7], [7+4, 6+6]]) = ½([[4, 11], [11, 12]]) = [[2, 5.5], [5.5, 6]]
3. Skew symmetric part K = ½(M – Mᵗ) = ½([[2-2, 4-7], [7-4, 6-6]]) = ½([[0, -3], [3, 0]]) = [[0, -1.5], [1.5, 0]]
4. So, M = S + K = [[2, 5.5], [5.5, 6]] + [[0, -1.5], [1.5, 0]] = [[2, 4], [7, 6]] ✅
This method is often asked in CBSE, JEE, and other competitive exam papers.
Try These Yourself
- Check whether matrix C = [[1, 8], [8, 2]] is symmetric or skew symmetric.
- Write the symmetric and skew symmetric parts of D = [[5, 6], [3, 4]].
- What is the sum of a symmetric and a skew symmetric matrix of same order?
- Is zero matrix symmetric, skew symmetric, or both?
- Show that in a 3x3 skew symmetric matrix, all diagonal elements are zero.
Frequent Errors and Misunderstandings
- Checking symmetry without confirming square shape (always check if matrix is square first).
- Forgetting that all diagonal elements in skew symmetric matrices must be zero.
- Mistakenly thinking all symmetric matrices are orthogonal (not true).
- Not using transpose correctly: swap rows for columns, not just rewrite.
Relation to Other Concepts
Learning symmetric and skew symmetric matrix strengthens your understanding of matrices, determinants, quadratic forms, and even applications in Physics (like moment of inertia). It also helps in understanding eigenvalues, as symmetric matrices always have real eigenvalues—a helpful point in higher maths exams!
Classroom Tip
A handy mnemonic: If a matrix “mirrors” perfectly across its diagonal, it’s symmetric (A = Aᵗ). If its mirror image is the reverse sign, it’s skew symmetric (A = -Aᵗ). Remember, in skew symmetric, diagonals are always zero! Vedantu teachers use tricks like “mirror or minus-mirror” to help students recall this faster.
Wrapping It All Up
Today we explored symmetric and skew symmetric matrix—their meaning, properties, step-by-step identification, tips, and exam tricks. These concepts connect to other key mathematical ideas and are useful in competitive exams and advanced science topics. Keep practicing with Vedantu Maths resources to strengthen your speed and accuracy with matrices.
Useful Tools and Resources
- Eigenvalue Calculator: Powerful for symmetric matrix properties in higher studies.
- Linear Programming: See how these concepts pop up in optimization problems.
FAQs on Understanding Symmetric and Skew Symmetric Matrices in Linear Algebra
1. What is a symmetric matrix?
A symmetric matrix is a square matrix that is equal to its transpose, meaning A = AT. This implies that the elements satisfy aij = aji for all i and j.
- It must be a square matrix.
- Elements across the main diagonal are equal.
- Example: If A = [[1, 2], [2, 3]], then AT = A, so it is symmetric.
2. What is a skew symmetric matrix?
A skew symmetric matrix is a square matrix whose transpose equals its negative, meaning AT = −A. This implies aij = −aji.
- It must be a square matrix.
- All diagonal elements are 0.
- Example: A = [[0, 3], [−3, 0]] is skew symmetric.
3. What is the condition for a matrix to be symmetric?
The condition for a matrix to be symmetric is A = AT. This means:
- The matrix must be square (same number of rows and columns).
- Each element satisfies aij = aji.
- The entries are mirror images across the main diagonal.
4. What is the condition for a matrix to be skew symmetric?
The condition for a matrix to be skew symmetric is AT = −A. This means:
- The matrix must be square.
- Each element satisfies aij = −aji.
- All diagonal elements must be 0 because aii = −aii.
5. Why are diagonal elements zero in a skew symmetric matrix?
The diagonal elements of a skew symmetric matrix are zero because aii = −aii, which implies aii = 0. Since each diagonal element must equal its own negative, the only possible value is zero.
6. What is the difference between symmetric and skew symmetric matrix?
The main difference is that a symmetric matrix satisfies A = AT, while a skew symmetric matrix satisfies AT = −A.
- Symmetric: aij = aji.
- Skew symmetric: aij = −aji.
- Diagonal entries: Any value (symmetric) vs 0 (skew symmetric).
7. How do you check if a matrix is symmetric or skew symmetric?
To check if a matrix is symmetric or skew symmetric, compare it with its transpose.
- Step 1: Find AT.
- Step 2: If A = AT, it is symmetric.
- Step 3: If AT = −A, it is skew symmetric.
- If neither condition holds, it is neither.
8. Can a matrix be both symmetric and skew symmetric?
A matrix can be both symmetric and skew symmetric only if it is the zero matrix. If A = AT and AT = −A, then A = −A, which implies A = 0.
9. Can every square matrix be written as the sum of a symmetric and a skew symmetric matrix?
Yes, every square matrix A can be expressed as the sum of a symmetric and a skew symmetric matrix using the formula:
- Symmetric part: (A + AT)/2
- Skew symmetric part: (A − AT)/2
- Thus, A = (A + AT)/2 + (A − AT)/2.
10. What are the properties of symmetric and skew symmetric matrices?
Symmetric and skew symmetric matrices have important algebraic properties in linear algebra.
- Symmetric matrix: A = AT, real eigenvalues, diagonalizable by orthogonal matrix.
- Skew symmetric matrix: AT = −A, diagonal entries are 0.
- The sum of two symmetric matrices is symmetric.
- The sum of two skew symmetric matrices is skew symmetric.
