Why must the determinant of a matrix with integer entries be an integer?
Note: I know what a determinant of a matrix is, not sure how to explain this question.
Is that because if the matrix is made with integers the determinant has to be an integer as well?
Best Answer
One of the various formulas for the determinant of an $n\times n$ matrix $A$ is $$\det(A) = \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma) \prod_{i=1}^n A_{i,\sigma_i} $$ where as usual $A_{i,j}$ denotes the entry of $A$ in the $i$th row and $j$th column, and $\mathrm{sgn}(\sigma)$ is either $+1$ or $-1$, depending on $\sigma$. Thus, if every $A_{i,j}$ is an integer, then $\det(A)$ is just a huge sum of products of integers, and is therefore an integer itself.