Symmetric and Skew Symmetric Matrices | Class 12 Maths

The knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. Now see one of the features of the matrix in this article. Here we discuss Symmetric and Skew Symmetric Matrices. According to matrices, only the square matrices can be symmetric or skew-symmetric form. Later in this article, we will discuss all things.

Symmetric Matrix

If for a matrix, the transposed form of that matrix is the same as the original matrix, then that matrix is said to be a Symmetric  Matrix. Let, a square matrix A of size n x n  is said to be symmetric if 

A^t = A

Where, [a_ij] = [a^_ji], for 1 ≤ i ≤ n, and 1 ≤ j ≤ n. In this case [a_ij] is an element at position (i, j) which is i^th row and j^th column in matrix A, and [a_ji] is an element at position (j, i) which is j^th row and i^th column in matrix A. So it is the transposed form of matrix A.

Example

So, this matrix is a Symmetric Matrix, because the transposed form of this matrix is itself the original matrix.

Skew-Symmetric Matrices

If for a matrix, the transposed form of that matrix is the same as the negative of the original matrix, then that matrix is said to be a Skew-Symmetric Matrix. Let, a square matrix A of size n x n  is  said to be skew-symmetric if

A^t = -A

Where, [a_ij] = [a^_ji], for 1 ≤ i ≤ n, and 1 ≤ j ≤ n. In this case [a_ij] is an element at position (i, j) which is i^th row and j^th column in matrix A, and [a_ji] is an element at position (j, i) which is j^th row and i^th column in matrix A. 

Example

Here, in the transposed form the matrix looks like the negative of the original matrix.

Properties of Symmetric and Skew-Symmetric Matrices

There are some rules that come from the concept of Symmetric and Skew-Symmetric Matrices,

1. If matrix A is a square matrix then (A + A^t) is always symmetric.

Prove: 

To find if a matrix symmetric or not, first, we have to find the transposed form of the given matrix 

So, let’s find the transpose of (A + A^t)

= (A + A^t)^t

= A^t + (A^t)^t

= A^t + A   [here, (A^t)^t = A]

= (A + A^t)

So, this is the same as the given matrix, so it is symmetric.

2. If matrix A is a square matrix then (A – A^t) is always skew-symmetric.

Prove:

To find if a matrix skew-symmetric or not, first, we have to find the transposed form of the given matrix 

So, let’s find the transpose of (A – A^t)

= (A − A^t)^t

= A^t − (A^t)^t

= A^t − A  [here, (A^t)^t = A]

= − (A − A^t) 

So, this form is the negative of the given matrix, so it is skew-symmetric.

Expressing Matrix in the form of Symmetric and Skew-Symmetric Matrices

Every square matrix can be expressed uniquely as the sum of symmetric and skew-symmetric matrices.

Prove:

Let A be a square matrix,

We can write, A = A/2 + A/2 

Let, A = P + Q

Now, find P^t and Q^t

and, 

So, here P is symmetric and Q is skew-symmetric matrices and A is the sum of P and Q.

Example

Express matrix A as the sum of a symmetric and skew-symmetric matrix, Where 

Answer: 

First, find the transpose of A

Now find (A + A^t) and (A – A^t)

Similarly:

Now, check the sum of (1/2)(A + A^t) and (1/2)(A – A^t) is the same as A or not,

So here A is expressed as the sum of the symmetric and skew-symmetric matrix.