How to find individual elements of a matrix?

Question

If I have a two-dimensional matrix, such that I know the sums of the rows and the columns, and that all the elements are in the set of integers [1..9], is there a simple way, other than guess-and-test, to determine each element? Let me set it up:

[a b]
[c d]

a + b = w, c + d = x, a + c = y, and b + d = z. a <> b, c <> d,
a <> c, and b <> d. If I know w, x, y, and z, how do I find a, b, c, and d?

And...go!

Answer 1

If you put thse equations into a matrix and solve them, you get:
a-d=y-x
b+d=x-y+w
c+d=x

d is a free variable here, so you still have to do some guessing, and even with the constraints that all the variables have different values, you might not get a unique solution. For instance, if w=10, x=7, y=8, and z=9, this is satisfied by both:
a = 2, b = 8, c = 6, d = 1, and
a = 3, b = 7, c = 5, d = 2

So without more information, there is no deterministic method of finding the values of a matrix from the row and column sums.

Answer 2

You can't.
You must have four equation to solve four unkown,
But you only have three equation.