Question 1. The goal of this exercise is to prove the following statement. (*) Let A and A' be 2 X 2 matrices. If the solution sets of the two homogeneous systems An? = (l and A's? = U are equal, then A and A' are row equivalent. The proof is divided into three steps. a) List all 2 x 2 matrices M that are in RREF. Remark. You do not need to explain why your list contains all such matrices. b) For every 2 x 2 matrix M in REEF, describe the solution set of the homogeneous system Ai" = 6'. Remark. You do not need to explain how you computed the solution set. Part a) and b) imply that the following intermediate result: (ant) Let M and M " be 2 x 2 matrices in REEF. If the solution set of the linear systems M 53' : fl and ME = 0 are equal, then M = M'. Remark. The proof of this intermediate result is not part of the exercises but you are strongly encouraged to think about how it follows from your computations. c) Combine results about RREFs and elementary row operations with (an) to prove (ale). Question 2. Let m, n 2 1 be natural number and consider a matrix A with m rows and n columns. This question concerns linear systems A5! = 5 with coefcient matrix A and augmented matrix [A E]. a) Assume 5 = 6 is the zero vector. Explain the relationship between the cardinality of the solution set of AL?! 2 0 and the number of free variables of A5 = 0. b) Assume that A has n = 2 columns and m > 2 rows. The goal of part b) is to show that under these conditions there always exists a vector 5 E Rm such that A5 2 his inconsistent. The proof is divided into two steps. b.1) Let M be the RREF of A. Find a vector 5\" E Rm such that M L? = I?" is inconsistent. b.2) Use part b.1) and results about elementary row operations to construct a vector 5 6 Rm such that A5 = 3 is inconsistent