What Is a Symmetric Matrix? Meaning, Formula & Solved Examples
Transpose of a Matrix
Let A be a matrix of order m x n; then the matrix of order n x m obtained by interchanging the rows and columns of A is called Transpose of the matrix A and is denoted by A’ or AT. For example, if
A = [fig 6] then A’= [fig 7]
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Note that A is a matrix of order 3x2 and its transpose A’ is a matrix of order 2x3.
Symmetric Matrix
A square matrix that is equal to its transpose is called a symmetric matrix. For example, a square matrix A = [aij] is symmetric if and only if aij= aji for all values of i and j, that is, if a12 = a21, a23 = a32, etc. Note that if A is a symmetric matrix then A’ = A where A’ is a transpose matrix of A.
Thus, A= [fig 4] is a symmetric matrix of order 3.
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Note that the transpose of A = A’ = [fig5] = A.
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How to Know If a Matrix is Symmetric
To know if the given matrix is symmetric or not, check the following conditions:
It should be a square matrix.
After transposing the matrix, it remains the same as that of the original matrix.
Symmetric Matrix Properties
The addition or subtraction of any two symmetric matrices will also be symmetric in nature.
The product of two symmetric matrices [A and B] doesn’t always give a symmetric matrix [AB]. The result of the product is symmetric only if two individual matrices commute (AB=BA).
The power on the symmetric matrix will also result in a symmetric matrix if the power n is integers.
The inverse of a symmetric matrix is also asymmetric.
Difference Between Symmetric and Skew-Symmetric Matrix
Determinant of Matrix
A fixed number that defines a square matrix is called the determinant of a matrix.
The process of finding the determinant of a symmetric matrix and the determinant of skew-symmetric is the same as that of a square matrix.
Matrix Inverse of a Symmetric Matrix
If A and B are two square matrices of the same order such that AB = BA = I, where I is the unit matrix of the same order as A. or B, then either B is called the inverse of A or A is called the inverse of B. The inverse of matrix A is denoted by A-1.
The inverse of a square matrix A exists if |A| is not equal to 0.
If A is nonsingular then, A-1 =
Let A and B are two nonsingular Matrices then,
i) A-1. A = A. A-1 = I
ii) (A-1)-1=A
iii) (A-1)T = (AT)-1
iv) (AB)-1= B-1A-1
FAQs on Symmetric Matrix in Maths: Key Concepts & Solved Questions
1. What is the definition of a symmetric matrix, with an example?
A symmetric matrix is a special type of square matrix that is equal to its own transpose. In simple terms, if you swap the rows and columns of the matrix, you get the exact same matrix back. For a matrix A to be symmetric, the condition is A = AT, which means the element in the i-th row and j-th column must be equal to the element in the j-th row and i-th column (aij = aji) for all i and j.
For example, the following 3x3 matrix is symmetric:
A =
| 1 7 3 |
| 7 4 -5 |
| 3 -5 6 |
Here, a12 = a21 = 7, a13 = a31 = 3, and a23 = a32 = -5.
2. How can you check if a matrix is symmetric?
To determine if a given matrix is symmetric, you must follow two simple steps as per the definition:
- Step 1: Check if it is a square matrix. A matrix must have the same number of rows and columns (e.g., 2x2, 3x3) to be symmetric. A non-square matrix cannot be symmetric.
- Step 2: Find its transpose. The transpose of a matrix is found by interchanging its rows with its columns.
- Step 3: Compare the transpose with the original matrix. If the transposed matrix is identical to the original matrix, then the matrix is symmetric.
3. Why can a matrix only be symmetric if it is a square matrix?
The fundamental condition for a matrix A to be symmetric is that it must be equal to its transpose (A = AT). For two matrices to be equal, they must have the same dimensions (order). If a matrix A has dimensions m x n, its transpose AT will have dimensions n x m. For A to equal AT, their dimensions must be identical, which means m x n must equal n x m. This is only possible if m = n. Therefore, only a square matrix can have the same dimensions as its transpose, making it the essential first condition for symmetry.
4. What is the key difference between a symmetric and a skew-symmetric matrix?
The key difference lies in their relationship with their transpose.
- A Symmetric Matrix is equal to its transpose (A = AT). The elements across the main diagonal are mirror images of each other (aij = aji).
- A Skew-Symmetric Matrix is equal to the negative of its transpose (A = -AT). This means the elements across the main diagonal are negative of each other (aij = -aji). A direct consequence is that all the elements on the main diagonal of a skew-symmetric matrix must be zero.
5. What are some important properties of symmetric matrices?
Symmetric matrices have several important properties that are useful in linear algebra:
- The sum and difference of two symmetric matrices are also symmetric.
- If A is a symmetric matrix, then its scalar multiple (kA) is also symmetric.
- The inverse of an invertible symmetric matrix (A-1) is also symmetric.
- The power of a symmetric matrix (An, where n is an integer) results in a symmetric matrix.
- For any square matrix A, the product A * AT and the sum A + AT are always symmetric matrices.
6. Is the product of two symmetric matrices always symmetric?
No, the product of two symmetric matrices is not always symmetric. Let A and B be two symmetric matrices. Their product, AB, is symmetric if and only if the matrices commute, which means AB = BA. If the matrices do not commute (AB ≠ BA), then the product AB will not be symmetric, even though A and B are symmetric individually.
7. How are symmetric matrices used in real-world applications?
Symmetric matrices are not just an abstract concept; they appear in many real-world applications across science and engineering. For example:
- In Statistics, covariance matrices are symmetric, representing the covariance between elements in a random vector.
- In Physics and Engineering, they are used to represent tensors, such as the stress tensor or the moment of inertia tensor, which describe physical properties of materials and systems.
- In Computer Science, the adjacency matrix of an undirected graph is always symmetric, as it represents a two-way relationship between nodes.
