Sample Problems on Symmetric and Skew Symmetric Matrices

Problem 1: Check whether the following matrix is symmetric or skew-symmetric.

[Tex]\bold{A = \begin{bmatrix} 2 & 5 & 8 \\ 5 & 1 & 7 \\ 8 & 7 & 4 \end{bmatrix}}[/Tex]

Solution:

As [Tex]A = \begin{bmatrix} 2 & 5 & 8 \\ 5 & 1 & 7 \\ 8 & 7 & 4 \end{bmatrix}[/Tex]

and [Tex]A^T = \begin{bmatrix} 2 & 5 & 8 \\ 5 & 1 & 7 \\ 8 & 7 & 4 \end{bmatrix} = A[/Tex]

Thus, the given matrix is symmetric matrix.

Problem 2: Is the following matrix symmetric?

[Tex]\bold{A = \begin{bmatrix} 5 & 2 & 1 \\ 2 & 3 & 4 \\ 1 & 4 & 6 \end{bmatrix}}[/Tex]

Solution:

As [Tex]A = \begin{bmatrix} 5 & 2 & 1 \\ 2 & 3 & 4 \\ 1 & 4 & 6 \end{bmatrix}[/Tex]

and Transpose of matrix A i.e., [Tex]A^T = \begin{bmatrix} 5 & 2 & 1 \\ 2 & 3 & 4 \\ 1 & 4 & 6 \end{bmatrix} = A[/Tex]

Thus, the given matrices is symmetric matrix.

Problem 3: Check whether the following matrix is symmetric or skew-symmetric.

[Tex]\bold{A = \begin{bmatrix} 0 & 4 & -7 \\ -4 & 0 & -3 \\ 7 & 3 & 0 \end{bmatrix}}[/Tex]

Solution:

As [Tex]A = \begin{bmatrix} 0 & 4 & -7 \\ -4 & 0 & -3 \\ 7 & 3 & 0 \end{bmatrix}[/Tex]

and [Tex]A^T = \begin{bmatrix} 0 & -4 & 7 \\ 4 & 0 & 3 \\ -7 & -3 & 0 \end{bmatrix} = -A[/Tex]

Thus, A given matrix is skew-symmetric matrix.

Problem 4: What type of matrices is the following matrix: symmetric or skew-symmetric?

[Tex]\bold{A = \begin{bmatrix} 0 & 3 & -7 \\ -3 & 0 & 2 \\ 7 & -2 & 0 \end{bmatrix}}[/Tex]

Solution:

As [Tex]A = \begin{bmatrix} 0 & 3 & -7 \\ -3 & 0 & 2 \\ 7 & -2 & 0 \end{bmatrix}[/Tex]

and [Tex]A^T = \begin{bmatrix} 0 & -3 & 7 \\ 3 & 0 & -2 \\ -7 & 2 & 0 \end{bmatrix} = -A[/Tex]

Thus, A given matrix is a skew-symmetric matrix.

Symmetric and Skew Symmetric Matrices are the types of square matrices based on the relation between a matrix and its transpose. These matrices are one of the most used matrices out of all the matrices out there. Symmetric matrices have use cases in optimization, physics, and statistics, whereas skew-symmetric matrices are used in subjects such as mechanics and electromagnetism. In this article, we will study the Symmetric and Skew Symmetric Matrices and their various properties such as eigenvalues and diagonalization.

The image below shows a symmetric and skew-symmetric matrix.