These are some good practice problems for anyone searching on the Web for determinants problems.
There is one or two questions that I am not getting right according to the system. Could you help me find which one? Ideally, I would like you to identify which one. Then, I would like to try to give my reasoning as to why I did it wrong.
My guess is that your answer to 8 is incorrect. This has little to do with your grasp of the material and everything to do with the ambiguous wording of the question.
I think that by "for all matrices $A$", they mean for arbitrary $n \times m$ matrices (that is, we are no-longer looking just at $n \times n$ matrices, only for the context of this question).