Here's what I would have submitted:


Question 1:

Rank is the number of pivots, so:
rk A=3,
rk B=3,
rk C=2,
rk D=3,
rk E=2,
rk H=rk J=3.



Question 2:

The pivot columns of H form a basis for its columnspace, so
[ 1]  [5]  [4]
[ 4]  [1]  [3]
[-1], [5], [3]
[-2]  [4]  [1]
[-2]  [2]  [3].

The pivot rows of J form a basis for the rowspace of H, so
[1 0 2 2 0], [0 1 -1 2 0], [0 0 0 0 1].



Question 3:

M is the change of basis from B1 to the standard basis, and N is the
change of basis from B2 to the standard basis.
We can change from B1 to B2 by first changing from B1 to the standard
basis, then changing from the standard basis to B2.

This is the same as multiplying first by M, then by N^-1, i.e.
multiplying by N^-1 M.

So the change of basis is N^-1 M.