We must understand what symmetric matrix is. 

A symmetric matrix is a square matrix that is equal to its transpose. 

A symmetric matrix ⇔ A = A^T 

Now, 

Let us understand what skew-symmetric matrix is.

A skew-symmetric matrix is a square matrix whose transpose equals its negative, that, it satisfies the condition 

A skew symmetric matrix ⇔ A T = - A 

And, 

A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. 

We need to find a square matrix which is both symmetric as well as skew symmetric. 

Take a 2 × 2 null matrix. 

Say,

A =\( \begin{bmatrix}0 &0 \\[0.3em]0 & 0 \\[0.3em]\end{bmatrix}\)

Let us take transpose of the matrix A. 

We know that, 

The transpose of a matrix is a new matrix whose rows are the columns of the original. 

So,

 A^T=\( \begin{bmatrix}0 &0 \\[0.3em]0 & 0 \\[0.3em]\end{bmatrix}\)

Since, 

A = A^T . 

∴ A is symmetric.

Take the same matrix and multiply it with -1.

 - A = -1 x\( \begin{bmatrix}0 &0 \\[0.3em]0 & 0 \\[0.3em]\end{bmatrix}\)

⇒ - A = -\( \begin{bmatrix}0 &0 \\[0.3em]0 & 0 \\[0.3em]\end{bmatrix}\)

 ⇒ - A = \( \begin{bmatrix}0 &0 \\[0.3em]0 & 0 \\[0.3em]\end{bmatrix}\)

Let us take transpose of the matrix –A. 

So,

  - A^T=\( \begin{bmatrix}0 &0 \\[0.3em]0 & 0 \\[0.3em]\end{bmatrix}\)

Since, 

AT = -A 

∴ A is skew-symmetric. 

Thus, 

A (a null matrix) is both symmetric as well as skew-symmetric.