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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?