    # Transpose Of A Matrix If A = [aij] be a matrix of order m x 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]mxn, then AT = [aji]mxn

Example: If A = , then AT = PROPERTIES OF TRANSPOSE OF THE MATRICES:

If A and B be two matrices of any suitable order then,
1. (AT)T = A

2. (A ± B)T = AT ± BT

3. (kA)T = kAT

4. (AB)T = BTAT
The above properties can be verified by examples which are given below:

1. Example:
2. Find the transpose of the matrix A = and verify that (AT)T = A.

Solution:

By interchanging the rows and columns of the matrix A we get the transpose of matrix A. Hence transpose of
matrix A = AT = .

Now (AT)T = = A

Hence, (AT)T = A

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

4. Solution:  Thus, (A + B)T = AT + BT

5. Example: If A = , then verify that (kA)T = kAT where k is any constant.

6. Solution:

Given that A = , then Then Hence (kA)T = kAT

7. Example: If and , verify that (AB)T = BTAT.

8. Solution: Hence BTAT = Hence (AB)T = BTAT

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