Do not use determinants

(1) (Oh, What Subsets Can Do!) Write down a set of vectors S Q R3 of size 5, such that o The largest linearly independent subset of S has size 2, and 0 el and 81+ 33 are in span(S). (2) (Geometric Series for Matrices) Suppose B is an n x n matrix such that Bk 2 0mm for some natural number In. Show that is an invertible matrix by writing down an explicit inverse A in terms of sums of powers of B. Verify that AB 2 BA 2 In. (3) (Solving Matrix Eqns. Is Hard) Write down a 2 x 2 matrix X such that O X2212 . X2202x2- Can these X s be the same matrix? Why or Why not? (4) (Matrices Like Being Multiplied In Order) Let A and B be in. x 5*}. matrices. Suppose that each column b, of B, for 1i 6 {1, 2,. . . ,n}, is in the nullspace of A. (a) Compute AB, and explain your reasoning