Symmetric Matrix and Skew-Symmetric Matrix

Symmetric Matrix

A square matrix A is said to be a symmetric matrix if   .

e.g. :  If    

                    
          Since   is a symmetric matrix.
Note : (1) In a symmetric matrix    for all i = j .
              (2) A rectangular matrix can never be symmetric .

Skew-Symmetric Matrix
A square matrix is said to be a Skew-symmetric matrix if  , i.e.  
for all i and j.

e.g. : If    

                       
But   

                       , which is the same as  .
Hence, A is a skew-symmetric matrix.
Note : In a skew-symmetric matrix   , for i = j
              i.e. all elements in the principal diagonal of a skew-symmetric matrix are zeroes.