Suppose we have an overdetermined matrix with $n$ rows and $n-1$ columns. We augment it with a column-matrix and then do Gaussian elimination on the augmented matrix. Assume that no row swapping is necessary, i.e., all pivot elements will be nonzero.
As a result of the Gaussian elimination, we obtain an upper-triangular matrix with $n$ rows and $n$ columns. Assume the last row is all-zero, except for its last element.
Is there a nice formula for expressing that last element in terms of the elements of the initial augmented matrix? In particular I'm interested in expressing it as a linear combination of the elements of the rightmost column of the initial augmented matrix.
My motivation is trying to avoid costly symbolic Gaussian elimination in a computer program, when all I want is the contents of that last element. This last element gives the constraint for the system of linear equations being consistent.
EDIT: it seems this isn't solvable as-is, but a nice formula does exist if we assume that the original unaugmented matrix is a Vandermonde matrix. Once we do a Laplace expansion on the last column of the augmented matrix, all the resulting minors are Vandermonde matrices. Vandermonde determinants are easy, so the problem is solved.