| # Permutation matrices A matrix is a permutation matrix if it is obtained by permuting the rows or columns of an identity matrix according to some permutation of the numbers to . Permutation matrices are orthogonal (hence, their inverse is their transpose: ) and satisfy . For example, the matrix is obtained by exchanging the columns and , and and , of the identity matrix. A permutation matrix allows to exchange rows or columns of another via the matrix-matrix product. For example, if we take any matrix , then (with defined above) is the matrix with columns and exchanged.