A permutation matrix is a matrix of zeros and ones for

Question

Question: A permutation matrix is a matrix of zeros and ones for which each row has one 1 and each column has one 1.

(a) Let B be an m × m matrix, and let P be a permutation matrix. Show that PB is a matrix obtained by permuting the rows of B and that BP is a matrix obtained by permuting the columns of B. Are the two permutations the same?

(b) Show that every permutation of the rows of a matrix B corresponds to multiplying B on the left by a permutation matrix.

(c) Show that for any permutation matrix P,

P^-1 = P^T.