Define the transpose of a matrix.

The transpose of a matrix is obtained by interchanging its rows and columns.

In linear algebra, a matrix is a rectangular array of numbers arranged in rows and columns. The transpose of a matrix is obtained by interchanging its rows and columns. For example, if A is a matrix with m rows and n columns, then the transpose of A, denoted by A^T, is a matrix with n rows and m columns. The element in the ith row and jth column of A^T is the same as the element in the jth row and ith column of A.

The transpose of a matrix has several important properties. Firstly, (A^T)^T = A, i.e., the transpose of the transpose of a matrix is the original matrix. Secondly, (AB)^T = B^T A^T, i.e., the transpose of a product of matrices is the product of their transposes in reverse order. Thirdly, (A + B)^T = A^T + B^T, i.e., the transpose of a sum of matrices is the sum of their transposes.

The transpose of a matrix is useful in many areas of mathematics and science, including linear algebra, calculus, and physics. It is used, for example, in solving systems of linear equations, finding eigenvalues and eigenvectors, and representing linear transformations.