The transpose of a matrix is denoted by is obtained by changing rows into columns or columns to rows of a matrix
. If size of the matrix
is
then the size of the transposed matrix
is
.
Transpose Of A Matrix
The element in row and
column of matrix
becomes the
row and
column element in matrix
.
Let
be a matrix of size
.
Transpose of matrix

Let us take element ‘c’ which is at 2nd row and 1st column of matrix ; after transpose operation on matrix
, it is at the position of 1st row and 2nd column of matrix
.
Similarly, the element ‘b’ is at the position of first row and second column of matrix , but after the transpose operation, its postion changes to 2nd row and 1st column in matrix
.
Example #1
Transpose the following matrix A.
The transpose of matrix
is

Example #2
Transpose the following matrix B.
The transpose of matrix
is

Symmetric Matrix
When the transpose of the matrix is the original matrix itself, then it is called a Symmetric matrix. Suppse is a matrix of size
, then the transpose of matrix
.
All the elements above the diagonal is a mirror image of elements below the diagonal elements. That is, is symmetric matrix if
for all i and j.
The elements of
. The transpose of such a matrix is,
Therefore,

What Are The Properties Of A Transpose Of A Matrix ?
In this section, we shall discuss about the properties of a transpose of a matrix. There are 4 interesting properties of a transpose as listed below.
, where
is a matrix of size
or
.
, where
and
are of same size, that is,
or
.
, where
is matrix of size
or
and
is a real number.
, where
and
are matrices of size
and
.
Let us verify each of the statement.
#1 : 
The transpose of a transpose of matrix is the original matrix
.
Let
Transpose of matrix
Transpose of

From the results above, it is clear that where
is a matrix of size
or
.
#2 : 
The transpose of sum of two matrices and
of same size
or
is equal to sum of transpose of matrices
and
.
Let
and
be two matrices of same size. Then
Transpose of matrix

Now, we shall take transpose of matrix and matrix
and add them together to obtain
.
Transpose of A
Transpose of B
Sum of
and

#3 : 
A transpose of the product of matrix with scalar
is equal to the product of scalar
and transpose of matrix
where size of the matrix
is
or
and
is a real number.
Let
and
Transpose of
Similarly, let us take transpose of
The product
is
Therefore, 
The output of both the products are equal and the property is true for all matrices.
#4 : 
The transpose of product of two defined ( and
) matrices
and
is equal to the product of transpose of matrix
and transpose of matrix
. Let us verify this claim with the help of an example.
Let
and
Transpose of
Similarly, the transpose of matrix
and matrix
is
and
Therefore, 
Once again, the product of both sides of the equation of the property holds true. The property is valid.
In the next, post we will discuss more about symmetric and skew-symmetric matrices.