If matrix P is a permutation matrix then P1PT ISolutionWe k

If matrix P is a permutation matrix, then P^-1P^T = I.

Solution

We know that a permutation matrix is a square matrix obtained from an identity matrix of the same dimension by a permutation of rows and that the inverse of a permutation matrix equals its transpose i.e. P^-1 = P^T .Then P^-1 P^T = (P^T)² . If P^-1 P^T = I, then (P^T)² = I so that P^T = ± I . Hence P = ± I which is a contradiction. Hence P^-1 P^T I.