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

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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 []. 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.