 # Symmetric Matrix A square matrix A = [aij] is said to be symmetric if A’ = A, that is, [aij] = [aji] for all possible values of i and j
For example: Since A = A’, A is symmetric.

## Skew - symmetric Matrix

A square matrix A = [aij] is said to be skew symmetric if A’ = -A, that is     aji = -aij 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.

Now try it yourself!  Should you still need any help, click here to schedule live online session with e Tutor!