abstract algebra?

Question

1)let M be a 3x3 matrix (2 1 1, 1 1 2, 0 2 1) (note that in this matrix row 1 is 2 1 1, row 2 is 1 1 2, ...). this matrix can be read as an element of Mat(3,F) for any field F, in particular for Q and for Z/p for every prime p. for which of these fields is M in GL(3,F)? for each field for which M is not invertible, find its null space.
2) consider the transpositions (12),(23),(34),(56),(67) in S_7. how can you prove that this is isomorphic to S_4XS_3, such that (12),(23),(34) generate S_4, and (56),(67) generate S_3.

Answer 1

1) det M = -5
so M is in GL(3,Q) and GL(3, Z_p) if p is not 5
To find the null space just solve the system
Mx = 0 in Z_5