Matrix Transpose Calculator

The transpose of a matrix A, written A to the T, swaps its rows and columns. The entry in row i, column j of A moves to row j, column i of A to the T. So an m by n matrix becomes an n by m matrix, and a column vector becomes a row vector.

Write the first row of A as the first column of the new matrix, the second row as the second column, and so on. Equivalently, reflect every entry across the main diagonal (the line from the top left to the bottom right). The calculator does this for any size from 1 by 1 up to 6 by 6.

A square matrix is symmetric when it equals its own transpose, so A[i][j] equals A[j][i] for every pair of indices. The entries form a mirror image across the main diagonal. Only square matrices can be symmetric, and this calculator flags the case for you.

A square matrix is skew-symmetric (also called antisymmetric) when its transpose equals its negative, so A[i][j] equals minus A[j][i]. That rule forces every entry on the main diagonal to be 0. The zero matrix is the only matrix that is both symmetric and skew-symmetric.

Yes. Transposition is an involution, which means (A to the T) to the T equals A. Swapping rows and columns once and then swapping them back leaves every entry in its original position, so you recover the matrix you started with.

If A has m rows and n columns, then A to the T has n rows and m columns. The two dimensions simply switch places. A 3 by 2 matrix transposes to a 2 by 3 matrix, and a square matrix keeps the same shape.