First we review some basic concepts of rank. Recall that elementary matrix operations do not change a matrix's rank. Let A R mn and B R np . Let In denote the n n identity matrix. (a) Perform elementary row and column operations1 to transform In 0 0 AB to B In 0 A . (b) Let's find lower and upper bounds on rank(AB). Use part (a) to prove that rank A + rank B n rank(AB). Then use what you know about the relationship between the column space (range) and/or row space of AB and the column/row spaces for A and B to argue that rank(AB) min{rank A, rank B}. (c) If a matrix A has rank r, then some r r submatrix M of A has a nonzero determinant. Use this fact to show the standard facts that the dimension of A's column space is at least r, and the dimension of A's row space is at least r. (Note: You will not receive credit for other ways of showing those same things.) (d) It is a fact that rank(A A) = rank A; here's one way to see that. We've