My question has 5 short parts and for each, i'm supposed to give a counterexample if the statement is false or give an argument to prove that a statement is true. 1.Q: Linear system with m equations and n variables. If m> or = to n, then the system can have at most 1 solution. A: I think this one is false because if the # of parameters= n-rank, since rank < or = n, if we take it to be less than n, then # of parameters = n - (some # less than n), which gives a # > 0, then this means there is at least 1 parameter and infinate solutions possible. 2.Q: A and B are matrices, and the product AB is defined. Then rank (AB) = rank A A: I would say that this is false because if A and B are different sizes, say A is 5x4 and B is 4x6, the rank of both A and B would have to be < or = 4, but AB produces a 5x6 matrix, meaning that the rank must be < or = 5. So couldn't A and B each have a rank of 4, while AB has a rank of 5? But if this is true, how can it be proved? 3.Q: A and B are nxn matricies and AB=0. Then at least one of A,B must have a determinant 0. 4.Q: An nxn matrix A satisfies AB=BA for every nxn matrix B, then A must be the identity matrix. I'm not sure about 3 or 4, any hints/points in the right direection would be appreciated. 5.Q: If the system Ax=b has no solution, then the system Ax=0 has only the trivial solution. A: I think this is false because the statements: The system Ax=0 has the only trivial solution x=0 and Ax=b has a unique solution for any vector b, according to the invertible matrix theorem. So since the bolded parts contradict each other, I assume its false but i'm not sure how to go about proving it. Any ideas?