Problem 6. Let P be an n × n permutation matrix with 1's on the anti-diagonal. Find det(P), Hint: How many exchange...
Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 < i S n 1 such that p, > Pi+1. How many permutations in Sn have exactly one descent?
Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 Pi+1. How many permutations in Sn have exactly one descent?
11. Let A= S- Find a matrix P such that P-IAP is a diagonal matrix.
Problem 2: Let pı .. .pn E Sn be a permutation, considered in its one-line notation. A descent in p is an index 1 < i S n 1 such that p, > Pi+1. How many permutations in Sn have exactly one descent?
A.
B.
(1 pt) 1 0 Let/ = 184 Find an invertible matrix P and a diagonal matrix D such that PDPA D= (1 pt) 1 5 -15 LetA=10-1 6 0-1 4 Find an invertible matrix P and a diagonal matrix D such that D = p- D=
(1 pt) 1 0 Let/ = 184 Find an invertible matrix P and a diagonal matrix D such that PDPA D=
(1 pt) 1 5 -15 LetA=10-1 6 0-1 4 Find an...
(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X-n -2? (c) What is the probability that X-n-1? (d) What is the expectation of X? (Hint:...
(1 point) Let A 12-5 Find an invertible matrix P and a diagonal matrix D such that D P- AP -24 12 5 D=
Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = −4 0 4 −3 −1 4 −6 0 6 , A5
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = p-1AP is the diagonal form of A. Prove that A* = Pokp-1, where k is a positive integer. Use the result above to find the indicated power of A. 10 18 A = -6 -11 18].46 A = 11
7.(6) Let A be a square matrix of size 4x4 and if det(A) = -1. Find det(3A) and the rank of A.
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = p-1AP is the diagonal form of A. Prove that Ak = pokp-1, where k is a positive integer. Use the result above to find the indicated power of A. -10 -18 A = 6 11 18].45 -253 -378 A6 = 126 188 11