Transpose of a matrix

Transpose of a matrix, or

The transpose of a matrix, written as , is obtained by interchanging the rows and columns of the original matrix A. Therefore if amd , the elemnt, of matrix equals to the element, of matrix, A. And if the order of A is m x n then the order of is n x m.

For example,

Properties of Matrix Transpose

, if and only if .

Matrix Products Transpose

if the matrix multiplication of exists, then .

Since is defined, is also defined while may not be.

Let , where A is of order m x n and B is of order n x r, and C is of order m x r then and element of C is

Let D is the transpose of C, then

That is element in the row j and colume i of matrix D equals to product of row i of matrix A and column j of matrix B, that is

Let , since is of order n x m and is of order r x n, then E is of order r x m.

Let X be the transpose of A, imply

,

Let Y be the transpose of B, imply

and imply

,

That is element in the row j and colume i of matrix E equals to

,

Subsitute corresponding elements in row of matrix B and in column of matrix A imply

,

And equals to the element of . Therefore

Transpose of Inverse Matrix

If exists, then . Since and , imply

,