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 defines a symmetric matrix, and can you provide an example from the CBSE Class 12 syllabus?
A square matrix 'A' is called a symmetric matrix if it is equal to its own transpose, meaning A = AT. In simpler terms, the element in the i-th row and j-th column is identical to the element in the j-th row and i-th column (aij = aji). For example, the 3x3 matrix A = [[1, 7, 3], [7, 4, -5], [3, -5, 6]] is symmetric.
2. What is a skew-symmetric matrix and what is its most distinct characteristic?
A square matrix 'A' is called a skew-symmetric matrix if it is the negative of its own transpose, meaning A = -AT. Its most distinct characteristic is that all its principal diagonal elements must be zero. Additionally, the elements across the diagonal are additive inverses of each other (aij = -aji). For example, B = [[0, -2, 3], [2, 0, -4], [-3, 4, 0]] is skew-symmetric.
3. What is the fundamental difference between a symmetric and a skew-symmetric matrix?
The fundamental difference lies in their relationship with their transpose and the properties that result from it:
- Symmetric Matrix: Satisfies the condition A = AT. The elements are mirrored across the main diagonal, and there are no restrictions on the diagonal elements.
- Skew-Symmetric Matrix: Satisfies the condition A = -AT. The elements across the main diagonal are negative of each other, and all diagonal elements must be zero.
4. Why must the diagonal elements of any skew-symmetric matrix always be zero?
This is a direct consequence of the definition. For any skew-symmetric matrix 'A', the condition is aij = -aji for all i and j. For any element on the main diagonal, the row and column indices are the same (i = j). Applying the rule, we get aii = -aii. If we bring the term to one side, we have 2aii = 0, which implies aii = 0. Therefore, every element on the main diagonal must be zero.
5. How can any square matrix be expressed as the sum of a symmetric and a skew-symmetric matrix?
As per a key theorem in the NCERT syllabus, any square matrix 'A' can be uniquely decomposed into the sum of a symmetric matrix (P) and a skew-symmetric matrix (Q). The formulas for this decomposition are:
- Symmetric part: P = ½ (A + AT)
- Skew-symmetric part: Q = ½ (A - AT)
6. Can a matrix be both symmetric and skew-symmetric at the same time?
Yes, but only the zero matrix (or null matrix) can be both. A matrix 'A' is symmetric if A = AT and skew-symmetric if A = -AT. For both conditions to hold true, it must be that A = -A. The only matrix that satisfies this equation is the zero matrix, where all elements are zero.
7. Why can't a non-square (rectangular) matrix be symmetric or skew-symmetric?
The concepts of symmetric and skew-symmetric matrices are defined only for square matrices. This is because the definition relies on comparing a matrix 'A' with its transpose 'AT'. If 'A' has dimensions m × n where m ≠ n, its transpose 'AT' will have dimensions n × m. Since the original matrix and its transpose have different orders, they cannot be equal, and thus the conditions A = AT or A = -AT cannot be met.
8. What are some practical applications of symmetric and skew-symmetric matrices?
These matrices are fundamental in various fields beyond textbook problems:
- Symmetric Matrices: Used in physics to represent the moment of inertia and stress tensors, in statistics for covariance matrices, and in computer science for graph theory (adjacency matrices of undirected graphs).
- Skew-Symmetric Matrices: Used in mechanics and robotics to describe rotational velocity and infinitesimal rotations. They are also essential in vector algebra for representing the cross product as a matrix multiplication.
9. How do you verify if a given square matrix is symmetric, skew-symmetric, or neither?
To verify the type of a given square matrix 'A', you must first calculate its transpose, AT. Then, you compare 'A' with 'AT':
- If A = AT, the matrix is symmetric.
- If A = -AT, the matrix is skew-symmetric.
- If neither of these conditions is met, the matrix is neither symmetric nor skew-symmetric.
