Operation of Transpose of a Matrix and its Properties
The transpose of a matrix is obtained by interchanging rows and columns of A and is denoted by AT.
More precisely, if A = [aij ]m×n , then AT = [bij ]n×m , where bij = a ji so that the (i, j)th entry of AT is a ji .
We state a few basic results on transpose whose proofs are straight forward.
For any two matrices A and B of suitable orders, we have
(i) ( AT )T = A
(ii) (kA)T = kAT (where k is any scalar)
(iii) ( A + B)T = AT + BT
(iv) ( AB)T = BT AT (reversal law on transpose)