It looks to me like B can be derived from A by "row operations".
(and a few "column" operations- but they work the same way.)
Do you know how row-column operations on a matrix affect its determinant?
Swapping two rows (columns) multiplies the determinant by -1.
Multiplying a row (column) by a number multiplies the determinant by that number.
Adding a multiple of one row (column) to another doesn't change the determinant.
Here's my two cents, assuming that I haven't made any arithmetic errors.
Put c=e=g=1, and the rest zero, and compute the determinants for the two matrices. I got |A|=-1, |B|=-12.
Put b=i=1, and the rest zero, and again compute the determinants for the two matrices. I got |A|= -2, |B|= -8.
Consequently, if you're looking for |B| to be a multiple of |A| you are going to be disappointed. So I think that the suggestions given so far for solving this problem will, as the OP says, come up short.
The restriction that |A|=1 is only one equation restricting 7 unknowns. This could get ugly.
the unknown values of the elements and its determinants are fixed values.
Of course everyone knows this already. Perhaps my post was too subtle.
Here's some guesses. You're taking a Linear Algebra class. You still don't have a solution to the problem. The instructor wrote the problem; it wasn't taken from a book. Instead, he made the problem up on his own (and thought that the answer was 12 or 44). Your instructor will admit the error sometime after the homework is due when some of the students complain that they can't work it either.
Possibly a misprint: if the lower left corner of B were 4h/2+ 2b, then it would be doable (and, unless I have misplaced an operation, the answer is -1/3). The instructer may have forgotten to divide the lower left corner by 3 when he (I think) multiplied the bottom row by 3.