Symmetric and Skew-Symmetric Matrices (Practice Problems)

A symmetric matrix and a skew-symmetric matrix are special types of square matrices defined by their relationship with their transpose.

• A symmetric matrix is a square matrix that is equal to its transpose.
• A skew-symmetric matrix is a square matrix whose transpose is the negative of the original matrix.

Sample Problems

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

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

Solution:

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

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

Thus, the given matrix is symmetric matrix.

Problem 2: Is the following matrix symmetric?

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

Solution:

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

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

Thus, the given matrices is symmetric matrix.

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

\bold{A = \begin{bmatrix} 0 & 4 & -7 \\ -4 & 0 & -3 \\ 7 & 3 & 0 \end{bmatrix}}

Solution:

As A = \begin{bmatrix} 0 & 4 & -7 \\ -4 & 0 & -3 \\ 7 & 3 & 0 \end{bmatrix}

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

Thus, A given matrix is skew-symmetric matrix.

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

\bold{A = \begin{bmatrix} 0 & 3 & -7 \\ -3 & 0 & 2 \\ 7 & -2 & 0 \end{bmatrix}}

Solution:

As A = \begin{bmatrix} 0 & 3 & -7 \\ -3 & 0 & 2 \\ 7 & -2 & 0 \end{bmatrix}

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

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

Practice Questions

Question 1: Determine whether the following matrix is symmetric, skew-symmetric, or neither.

{A = \begin{bmatrix} 4 & 2 & -1 \\ 2 & 5 & 3 \\ -1 & 3 & 6 \end{bmatrix}}

Question 2: Check whether the following matrix is symmetric, skew-symmetric, or neither.

\bold{A = \begin{bmatrix} 0 & 6 & -2 \\ -6 & 0 & 5 \\ 2 & -5 & 0 \end{bmatrix}}

Question 3: Identify the type of the following matrix.

\bold{A = \begin{bmatrix} 1 & 4 & 2 \\ 4 & 3 & -5 \\ 3 & -5 & 2 \end{bmatrix}}

Question 4: Determine whether the following matrix is symmetric, skew-symmetric, or neither.

{A = \begin{bmatrix} 0 & -3 & 4 \\ 3 & 0 & 1 \\ -5 & -1 & 0 \end{bmatrix}}