Symmetric Matrix: Overview, Questions, Preparation

A symmetric matrix is a type of matrix that is equal to its transpose; a symmetric matrix is always a square matrix. Consider a square matrix A; A is a symmetric matrix, if A = AT. 

Properties of Symmetric Matrix

If we add or subtract two symmetric matrices, the resultant matrix will be symmetric.

If a matrix P is symmetric, then Pn will also be a symmetric matrix, where n is an integer. In this case, P-1 will also be a symmetric matrix.

If two symmetric matrices have commutative properties, then the product of these two matrices is also symmetric.

A =

[1 2 3]

[2 6 7]

[3 7 8]

Transpose of A (AT) =    [1 2 3]

                                      [2 6 7]

                                      [3 7 8]

Here A = AT
Hence, A is a symmetric matrix.

How to Check If a Matrix Is Symmetric?

Step 1: Take a transpose of the matrix.

Step 2: Compare the matrix with the transpose.

Step 3: If the matrix is equal to its transpose, then the matrix is symmetric.

Skew-Symmetric Matrix

A matrix is a skew-symmetric matrix if its transpose is equal to the negative of the original matrix. If A is a matrix having its transpose as AT, then A is a skew matrix if AT = -A.
Transpose of a Matrix

The transpose of a matrix is determined by interchanging the rows and columns of the matrix. It implies that the rows in the matrix become columns and the columns become rows in its transpose.

For example matrix A = [11 12 13 14]
                                      [15 16 17 18]

Transpose of A (AT) = [11 15]
                                    [12 16]
                                    [13 17]
                                    [14 18]

Symmetric matrices are the basis of matrices, and it is necessary to understand the topic to study other advanced topics of matrices and various operations based on it. Symmetric matrices are part of the Matrices chapter of the Class XII Maths syllabus. It is important to understand the basic concept of matrices, symmetric matrices, and operations performed on matrices. Students will get 1 question based on symmetric matrices in the examination.

1. What is the transpose of the below matrix:
[1 -2]
[2 3]

Solution.  Transpose (AT) = [1   2]
                                              [-2  3]

2.Verify that for below matrix A, (A+AT) is symmetric matrix:
[1 5]
[6 7]

Solution.

 A = [1 5]
       [6 7]

AT = [1 6]
        [5 7]

A+AT = [1 5] + [1 6] = [2 11]

[6 7]     [5 7] [11 14]

 (A+AT) T = [2 11]

                  [11 14]

A+AT = (A+AT) T
Hence, A+AT is a Symmetric matrix.

3.Verify if below matrix A is symmetric:
[1 5]
[-1 2]

Solution.

 AT = [1 -1]
         [5 2]
Here, A != AT. Hence, A is not a symmetric matrix.

Q: How can we check if any matrix is a symmetric matrix?

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

Q: Which matrix is symmetric but is non-invertible?

Q: Is zero matrix symmetric matrix?

Q: What is an orthogonal matrix?