# Transpose of a Matrix

## Transpose of a Matrix

Let A be a matrix. Then a new matrix obtained by interchanging the corresponding rows and columns of A is called the transpose of A. It is denoted by A’ or At.

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******************** Here, the order of matrix A is 2 × 3 and that of At is 3 × 2. Hence if the order of a matrix A is m × n, then the order of transpose of matrix A i.e. At will be n × m. If A is a square matrix of order n, then its transpose At is also a square matrix of order n. If A is a row matrix, then its transpose At is a column matrix.

For example: ### Properties of Transpose of a Matrix

1.   The transpose of the transpose of a matrix is the matrix itself, i.e. (At)t = A. 2.   The transpose of the sum of two matrices is equal to the sum of their transposes, i.e. (A + B)t = At + Bt. 3.   If A is any matrix and k is any number, then (kA)t = kAt. 4.   If A and B are two matrices conformable for multiplication, then (AB)t = BtAt. ### Worked Out Examples   #### Do you have any questions regarding the transpose of a matrix?

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