# Symmetric Matrix

A square matrix A = [a_{ij}] is said to be symmetric if A’ = A, that is, [a_{ij}] = [a_{ji}] for all possible values of i and j

For example:

Since A = A’, A is symmetric.

## Skew - symmetric Matrix

A square matrix A = [a_{ij}] is said to be skew symmetric if A’ = -A, that is a_{ji} = -a_{ij} for all possible values of i and j.

For example:

Here B = B’, so it is skew symmetric.

## Important Results

1) For any square matrix A with real number entries, A + A’ is a symmetric matrix and A - A’ is a skew symmetric matrix.

Example: For the matrix verify that

(i) (A + A’) is symmetric matrix

(ii) (A - A’) is skew symmetric matrix

Solution:

From (a) and (b),

(A + A’) = (A + A’)’, so it is symmetric matrix

From (c) and (d),

(A - A’) = -(A - A’)’, so it is skew symmetric matrix.

2) Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.

Example: Express

As the sum of symmetric and a skew- symmetric matrices

Solution:

Let P = ½ [A + A’] and Q = ½ [A - A’]. We have to show that A = P + Q

Hence P + Q = A, so A is expressed as sum of a symmetric and skew symmetric matrices.

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