TRANSPOSE OF A MATRIX

Definition

Let A = [ a_ij] be an m×n matrix .Then , the transpose of A , denoted by A^T or A' , is an n×m matrix such that

 (A')_ij = a_ji for all i =1,2,...,m ;j =1,2,...,n.

Thus, A' is obtained from A by changing its rows into columns and its columns into rows.

Properties of Transpose

(1)For any matrix A , (A')'=A

(2)For any two matrices A and B of the same order ,

      (A + B )' = A' +B'

(3) IF A is a matrix and k is a scalar , then

       (kA)' = k(A')

(4)If A and B are two matrices such that AB is defined , then

      (AB)' = B' A'

Symmetric and Skew -Symmetric Matrix

Symmetric Matrix

A square matrix A =[a_ij] is called a symmetric matrix , if

         a_ij =a_ji    for all i , j.

Or      A = A'

For example -