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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
    5.
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

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

    3. Solution:





    Thus, (A + B)T = AT + BT

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

    4. Solution:

    Given that A = , then



    Then

    Hence (kA)T = kAT

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

    5. Solution:



    Hence

    BTAT =

    Hence (AB)T = BTAT


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