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SYMMETRIC AND SKEW SYMMETRIC MATRICES

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Symmetric Matrix

symmetricandskew1

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:  symmetricandskew2
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: symmetricandskew3
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 symmetricandskew4    verify that
(i) (A + A’) is symmetric matrix
(ii) (A - A’) is skew symmetric matrix
Solution:
symmetricandskew5
From (a) and (b),
(A + A’) = (A + A’)’, so it is symmetric matrix

symmetricandskew6
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 symmetricandskew7
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

symmetricandskew8
Hence P + Q = A, so A is expressed as sum of a symmetric and skew symmetric matrices.                 

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