Say you have a 4x3 matrix P, whose entries are all integers.
What is a necessary and sufficient condition on such that (where and ) has a solution?
If you write this out as a linear system it is overdetermined. It is saying that you have 4 equations in 3 unknowns. So a necessary condition would be that two of the rows (in the augmented) matrix must be identical. This alone is not enough sufficient. Now if you eliminate this 4th row from both you will now have a system of equations in 3 variables with 3 unknowns. Now what has to be true for this system to have a solution for the new column vector .