TITLE: Do the matrices with maximum determinant always have integral values
QUESTION [12 upvotes]: Consider all $n$ by $n$ matrices whose elements are real values in $[0,1]$.  Now for a given $n$, consider all those matrices with maximum determinant.

Are all the elements of these matrices always either exactly $0$ or $1$?

From numerical experiments this seems to be the case but I don't see how to prove it.

A very nice counterexample was given to my question as I didn't phrase it correctly. It should have been:

Can the maximum determinant always be reached by a matrix with only
  $1$ and $0$ entries?

REPLY [5 votes]: I should have asked, can the maximum always be achieved by a matrix with only 1 and 0 entries?

Yet another way to see that the answer to this question is "yes" is to note that the determinant is a multilinear function of the columns (or, equivalently, the rows) of the matrix.  In particular, this implies that the determinant is also a multilinear function of the individual matrix elements — i.e. that, if we hold all the other elements constant and vary only one element $a_{ij}$ of the matrix, the determinant changes linearly as a function of that element.
Thus, if our matrix has any element strictly between 0 and 1, we can always set that element to either 0 or 1 without decreasing the determinant.  By repeating this process for each element of the matrix, we obtain, for any initial matrix $A \in [0,1]^{n \times n}$, another matrix $A' \in \{0,1\}^{n \times n}$ such that $|A| \le |A'|$.  In particular, if the determinant of $A$ is maximal, so is that of $A'$.