You are given an n n matrix A with rows r1, . , rn. Let B be the matrix with rows rn, rn1, . , r1. Prove that det(B) = (1)( n(n1))/ 2 det(A). [Hints:...

You are given an n × n matrix A with rows r1, . . . , rn. Let B be the matrix with rows rn, rn−1, . . . , r1. Prove that det(B) = (−1)( n(n−1))/ 2 det(A). [Hints: Use elementary row operations, only swapping adjacent rows. To get a feel for the 2 argument first prove the statement for n = 2, 3, 4. Generalising to all n is only a matter of visualising (by drawing a series of pictures) and counting.]