**A symmetric matrix is a matrix of order n with the same number of rows and columns where its transposed matrix is equal to the original matrix.**

In other words, a symmetric matrix is a square matrix and is identical to the matrix after having swapped rows for columns and columns for rows.

## Requirements

For any matrix to be a symmetric matrix, it must meet the following restrictions:

Given a symmetric matrix **P** of order n,

- Be a
**square matrix**.

The number of rows (n) has to be the same as the number of columns (m). That is, the order of the matrix has to be n given that n = m.

- The original matrix has to be equal to its
**transposed matrix**.

Demonstration:

## Properties

- The adjoint matrix of a symmetric matrix is also a symmetric matrix.

Demonstration:

- The addition or subtraction of two symmetric matrices results in another symmetric matrix.

Demonstration:

Given two symmetric matrices **P** and **T** of order 3, we obtain another symmetric matrix **S** from the sum.

## Why is it called a symmetric matrix?

The property of symmetry is given by the elements around the main diagonal. Since a square matrix is a symmetric matrix, it will always have the same number of elements above and below the main diagonal. These elements are the same symmetrically. That is, the main diagonal acts as a mirror.

## Proof of symmetry and skewness of a matrix

## Symmetric matrix

The letter **d** represents the elements of the main diagonal. The other letters represent any real number. We can see that the main diagonal acts like a mirror: it reflects the elements on both sides. In other words, when the elements on both sides of the diagonal are symmetrically equal, we say that the matrix **P** is a symmetric matrix.

## Non-symmetric matrix

Matrix **X** is not a symmetric matrix since it is not a square matrix and its transposed matrix is different from the original matrix. In addition, it does not have a main diagonal either.

Non-symmetric matrix