Is the determinant of a matrix different depending on the field over which it is viewed

Question

General question:

Suppose we have a square matrix over R. Will its determinant change if we view the same matrix over a finite field (i.e. Z\5Z) or not?

Answer 1

I am not sure how about general entries from $R$. Surely if the other field is containing $R$ as a subfield then nothing changes. If you are considering different fields you need entries in matrix $A$ to be in this different field (ie. in the intersection of $R$ and this field). I guess that to make it workable you need both of these fields to be contained in a bigger field.

In a less general setting, this is indeed true. Take for example matrix with only zeroes and ones (e.g. incidence matrix of some graph or set system). Then by viewing it in a different field (real numbers, $F_2$, $F_p$ for $p$ prime) you can get different determinants:

A determinant is some sum over all permutations. These summands can be either zero or one since $A$ is 0-1 matrix (or -1 if the sign of the permutation is negative and fieůd is different from $F_2$).

For example, if determinant results to be $2$ for a matrix understood as a real matrix, over field $F_2$ you get determinant $0$ and over field $F_3$ you still get $Det(A)=2$.

When applying methods of Linear Algebra in solving problems from combinatorics it can be indeed useful to understand incidence matrices over a different field (especially when asking whether it is a regular matrix).