Why must the determinant of a matrix with integer entries be an integer?

Question

My answer would be because it involves only addition, subtractions and multiplications, no divisions. But I am not 100%sure. 

Answer 1

Another way to see this is the fact that a Determinant of a matrix is simply a linear combination of the entries (all of them in fact)...When we restrict the elements of the matrix to the Integers, we can make use of the fact that Z={all integers} form what is called a Ring.  Namely it is closed under the operations of addition and multiplication...which means that all linear combinations of elements in Z yield another element of Z!

 

Thus, since the determinant of a matrix with integer values is a linear combination of integers, it must also be an integer.

Answer 2

The formal definition of the determinant of an n x n matrix A is as a sum n! products where each product is that of n of A's entries times ±1 [http://en.wikipedia.org/wiki/Determinant#n-by-n_matrices]. Assuming all entries are integers, each product must be an integer and in turns, the sum of those products is an integer because integers are closed under addition and multiplication.

Don't worry about the notation used in wikipedia.

Similarly, if the matrix has entries all from a set S that is closed under + and *, then the determinant is in S as well.