We examine relationships between two minors of order n of some matrices of n rows and n + r columns. This is done through a class of determinants, here called n-determinants, the investigation of which is our objective.

As a consequence of our main result we obtain a generalization of theorem of the product of two determinants.

We show the upper Hessenberg determinants, with -1 on the subdiagonal, belong to our class. Using such determinants allow us to represent terms of various recurrence sequences in the form of determinants. We illustrate this with several examples. In particular, we state a few determinants, each of which equals a Fibonacci number.

Also, several relationships among terms of sequences defined by the same recurrence equation are derived.