Test 2, Q1:

You are supposed to prove here that for any matrices A,B. Why, then, do you assume that B is singular?

Test 2, Q2:

You are meant to give a counter example here, like you did in Q3.

Test 2, Q4:

Your proof is incorrect. You are saying, essentially, that if A,B are row equivalent then where but then and clearly not any two row equivalent matrices are equal. The correct formulation is that with elementary matrices.

You should also note that this proposition is not true: Take and do the row operation to get B, then:

Test 2, Q5:

This is also incorrect. I don't see how supposedly not having a third coordinate has anything to do with closure under addition. You should note that in , would be . It should now be obvious thatisa subspace of (and this makes much more sense, does it not?)

Test 4, Q3:

As already mentioned, if you think a proposition is false you need to provide a counter example, not just say that it may not always be true.

Test 4, Q5:

0 is not a positive number.