Let G be a subset of Msub3(R) satisfying the following three conditions:
1. For every A and B in G, the matrix product AB is in G
2. There is a matrix E in G such that EA = A = AE for all A in G
3. For every A in G, there exists a B in G such that AB = E = BA
I'm supposed to figure out if the following statements are true or false. Prove our counterexample them.
1. E is the 3 x 3 identity matrix.
2. If A is in G then Det(A) is not equal to zero.
My feeling is that 1 is true, but I don't know how to prove it. And I'm not sure about 2.