## Transpose of a Matrix

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# Transpose of a Matrix

Definition. Let A be an mxn matrix. The transpose of A, denoted by A or AT, is the n x m matrix obtained from A by interchanging the rows and columns of A. Thus the first row of A is the first column of A the second row of A is the second column of A and so on.

1. Let A and B be two matrices of order m x n. Then (A ± B )' = A' ± B'.

2. Let A be a matrix of order m x n and k be a scalar. Then (Ka) = KA’.

3. Let A and B be matrices of order m x n and n x p respectively. Then (AB)’ = B’ A’.

4. For any matrix A, (A’) = A.

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**Example**. If A = , find A.**Solution.**Column 1 of A becomes Row 1 of A Column 2 becomes Row 2 and Column 3 becomes Row 3. Thus**Properties of the Transpose of a Matrix**1. Let A and B be two matrices of order m x n. Then (A ± B )' = A' ± B'.

2. Let A be a matrix of order m x n and k be a scalar. Then (Ka) = KA’.

3. Let A and B be matrices of order m x n and n x p respectively. Then (AB)’ = B’ A’.

4. For any matrix A, (A’) = A.

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