the identity matrix, and two invertible matrices.

1)

a) What about ?

b) .

The first equality is true because is a ring.

c) What about and associativity of multiplication.

d) , so

e)Try with and

f) What we can write in any ring is

You get the asked equality iff , that is to say iff . So e) answers the question.

2)You just have to check how many vectors are independant in each set. Note that a subspace of has a maximal dimension of , so if a set of vectors contains at least independant vectors, then