An n n matrix is said to be a permutation matrix if every row and every column has exactly one

non-zero entry, and this entry is a 1.

(a) If T : Rⁿ Rⁿ has a matrix form in some basis that is a permutation matrix, describe what T is doing to this basis.

(b) if P and Q are two permutation matrices, show that P Q is as well. Describe this matrix?

(c) Prove that any permutation matrix must be invertible, and that its inverse is also a permutation

matrix. Describe the inverse matrix?

(d) Show that there must exist some N so P^N is the identity matrix. (Do not do this by multiplying permutation matrices.)