How to Check if a Matrix is Symmetric or Skew-Symmetric?
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 Symmetric Matrix and Skew-Symmetric Matrix Explained with Examples
1. What is a symmetric matrix?
A symmetric matrix is a square matrix that is equal to its own transpose. This means that if A is a symmetric matrix, then A = AT, where AT represents the transpose of matrix A. In simpler terms, the element in the ith row and jth column is equal to the element in the jth row and ith column. The diagonal elements can be any value.
2. What is a skew-symmetric matrix?
A skew-symmetric matrix (also called an antisymmetric matrix) is a square matrix whose transpose is equal to its negative. If A is a skew-symmetric matrix, then A = -AT. This implies that the elements on the main diagonal are always zero (since aii = -aii only if aii = 0), and elements symmetrically opposite the main diagonal are opposites of each other (aij = -aji).
3. What is the difference between a symmetric and a skew-symmetric matrix?
The key difference lies in the relationship between the matrix and its transpose. A symmetric matrix satisfies A = AT, meaning it's equal to its transpose. A skew-symmetric matrix satisfies A = -AT, meaning it's equal to the negative of its transpose. Consequently, the diagonal elements of a skew-symmetric matrix are always zero.
4. How do you check if a matrix is symmetric?
To check if a matrix is symmetric, find its transpose (AT). If A = AT, then the matrix is symmetric. This means you compare each element aij with the corresponding element aji. If they are equal for all i and j, the matrix is symmetric.
5. How do you check if a matrix is skew-symmetric?
To check if a matrix is skew-symmetric, find its transpose (AT). If A = -AT, then the matrix is skew-symmetric. This means you compare each element aij with the corresponding element aji. If aij = -aji for all i and j (and aii = 0 for all i), the matrix is skew-symmetric.
6. Can a matrix be both symmetric and skew-symmetric?
Yes, but only the zero matrix (a matrix where all entries are zero) can be both symmetric and skew-symmetric. This is because the only matrix that satisfies both A = AT and A = -AT is the zero matrix.
7. What is the sum of a symmetric and a skew-symmetric matrix?
Any square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. This decomposition is unique. Let A be any square matrix. Then A can be written as A = B + C, where B = (A + AT)/2 is a symmetric matrix, and C = (A - AT)/2 is a skew-symmetric matrix.
8. Give an example of a 2x2 symmetric matrix.
A simple example is: [[1, 2], [2, 3]]. Note that the off-diagonal elements (2) are equal.
9. Give an example of a 2x2 skew-symmetric matrix.
A simple example is: [[0, 2], [-2, 0]]. Note that the off-diagonal elements are opposites, and the diagonal elements are zero.
10. What are some applications of symmetric and skew-symmetric matrices?
Symmetric matrices are used extensively in various fields, including representing quadratic forms, solving systems of linear equations, and in physics (e.g., moment of inertia tensors). Skew-symmetric matrices find applications in representing rotations and cross products in vector spaces and in various aspects of physics and engineering.
11. Are the eigenvalues of a real symmetric matrix always real?
Yes, the eigenvalues of a real symmetric matrix are always real numbers. This is a crucial property in many applications.
