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Properties of Transposes

If A = |aij| be a matrix of order m × n, then the matrix obtained by interchanging the rows and columns of A is known as the transpose of A. It is represented by AT.

Hence if A = |aij| of order m × n, then AT= |aij| of order n × m.


Example: If , then

The following properties are valid for the transpose:
·  The transpose of the transpose of a matrix is the matrix itself: (AT)T = A
·  Transpose of a scalar multiple: The transpose of a matrix times a scalar (k) is equal to the constant times the transpose of the matrix: (kA)T = kAT
·  Transpose of a sum: The transpose of the sum of two matrices is equivalent to the sum of their transposes:
(A + B)T = AT + BT
·  Transpose of a product: The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order: (AB)T = BT AT 
·  The same is true for the product of multiple matrices: (ABC)T = CTBTAT .


Example 1: Find the transpose of the matrix  and verify that (AT)T = A.
Solution:
The transpose of matrix A is determined as shown below:


And the transpose of the transpose matrix is:


Hence (AT)T = A.

Example 2: If   and , verify that (A ± B)T = AT ± BT.
Solution:

and the transpose of the sum is:

The transpose matrices for A and B are given as below:

And the sum of the transpose matrices is:

Hence (A ± B)T = AT ± BT.


Example 3: If  and , verify that (AB)T = BT AT .

Solution:
The product of A and B is:


And the transpose of (AB) is:


If we take the transpose of A and B separately and multiply A with B, then we have:


Hence (AB)T = BT AT .
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