The transpose of a matrix A, denoted AT, is an operation that interchanges the corresponding rows and columns of a matrix:


Specifically, transposing a matrix is an operation that changes the position of each index in a matrix (refer to the matrix notation page for a refresher if necessary), such that an n x m matrix becomes an m by n matrix, making the (i,j)th entry of the matrix the (j,i)th. For example, in matrix A above, entry a11, becomes entry a21, which in both cases is 1. Similarly, entry a23 becomes entry a32, such that the "6" is transposed from the 3rd column of the 2nd row, to the 2nd column of the 3rd row in the transposed matrix. The general form of a matrix and its transpose can be written as follows:

Note that in the transposed matrix AT, the indices no longer indicate the corresponding entry in the matrix. Rather, it shows that the first row in matrix A becomes the first column in the transposed matrix.

Properties of transposition

Below is a list of properties of matrices that involve transposition, as well as examples of each:

  1. , for some scalar k
  2. , where A is a square matrix
  3. , where A is a square invertible matrix

i. For ,


ii.For k = 3 and ,

iii. For and


iv. For For and ,




v. For , where "det" indicates taking the determinant of the matrix,




vi. For , where A-1 indicates the inverse of matrix A,