Transpose of a Matrix

Transpose of a matrix is the matrix obtained by changing its rows into corresponding columns or columns into corresponding rows. The transpose of a m × n matrix is a n × m matrix that results by interchanging the rows and columns of the matrix. The transpose of a matrix is written with the superscript T or ‘.

 

 

 

Example,

 

 

 

Properties

• The transpose of the transpose of a matrix is the matrix itself.

(A^T)^T= A

• The transpose of a matrix times a scalar is equal to the scalar times the transpose of the matrix.

(kA)^T = kA^T

• The transpose of the sum of two matrices is equivalent to the sum of their transposes.

(A + B)^T = A^T + B^T

The transpose of the product of two matrices is equivalent to the product of their transposes in reversed.

(A × B)^T = B^T × A^T

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