TITLE: What is the maximum possible value of determinant of a matrix whose entries either 0 or 1?
QUESTION [12 upvotes]: My question is simply the title:

What is the maximum possible value of determinant of a matrix whose entries either 0 or 1 ?

REPLY [3 votes]: A few examples of $\{0,1\}$-matrices (with the largest determinants $-$ according to OEIS-A003432):
$n=2$:
$\quad\det\left(
\begin{array}{cc}
\bf{1} & 0 \\
\bf{1} & \bf{1} \\
\end{array}
\right)
=1;$
$n=3$:
$\quad\det\left(
\begin{array}{ccc}
\bf{1} & 0 & \bf{1} \\
\bf{1} & \bf{1} & 0 \\
0 & \bf{1} & \bf{1} \\
\end{array}
\right)
=2;$
$n=4$:
$\quad\det\left(
\begin{array}{cccc}
\bf{1} & 0 & \bf{1} & 0 \\
\bf{1} & \bf{1} & 0 & \bf{1} \\
0 & \bf{1} & \bf{1} & 0 \\
0 & 0& \bf{1} & \bf{1} \\
\end{array}
\right)
=3;$
$n=5$:
$\quad\det\left(
\begin{array}{ccccc}
\bf{1} & 0 & \bf{1} & 0 & 0\\
\bf{1} & \bf{1} & 0 & \bf{1} & 0 \\
0 & \bf{1} & \bf{1} & 0 &\bf{1}\\
0 & 0 & \bf{1} & \bf{1} & 0 \\
\bf{1} & 0 & 0 & \bf{1} & \bf{1} \\
\end{array}
\right)
=5;$
$n=6$:
$\quad\det\left(
\begin{array}{ccccc}
\bf{1} & 0 & \bf{1} & 0 & 0 & 0\\
\bf{1} & \bf{1} & 0 & \bf{1} & 0 & 0 \\
0 & \bf{1} & \bf{1} & 0 &\bf{1} & 0\\
0 & 0 & \bf{1} & \bf{1} & 0 & \bf{1}\\
\bf{1} & 0 & 0 & \bf{1} & \bf{1} & 0\\
\bf{1} & \bf{1} & 0 & 0 & \bf{1} & \bf{1} \\
\end{array}
\right)
=9;$
$n=7$:
$\quad\det\left(
\begin{array}{ccccc}
\bf1 &    0 & \bf1 &    0 &    0 & \bf1 & \bf1 \\
\bf1 & \bf1 &    0 & \bf1 &    0 &    0 & \bf1 \\
\bf1 & \bf1 & \bf1 &    0 & \bf1 &    0 &    0 \\
   0 & \bf1 & \bf1 & \bf1 &    0 & \bf1 &    0 \\
   0 &    0 & \bf1 & \bf1 & \bf1 &    0 & \bf1 \\
\bf1 &    0 &    0 & \bf1 & \bf1 & \bf1 &    0 \\   
   0 & \bf1 &    0 &    0 & \bf1 & \bf1 & \bf1 \\
\end{array}
\right)
=32;$
$n=8$:
$\quad\det\left(
\begin{array}
\bf1 &    0 & \bf1 &    0 &    0 & \bf1 & \bf1 &    0 \\
\bf1 & \bf1 &    0 & \bf1 &    0 &    0 & \bf1 & \bf1 \\
\bf1 & \bf1 & \bf1 &    0 & \bf1 &    0 &    0 & \bf1 \\
   0 & \bf1 & \bf1 & \bf1 &    0 & \bf1 &    0 &    0 \\
   0 &    0 & \bf1 & \bf1 & \bf1 &    0 & \bf1 &    0 \\
\bf1 &    0 &    0 & \bf1 & \bf1 & \bf1 &    0 & \bf1 \\
   0 & \bf1 &    0 &    0 & \bf1 & \bf1 & \bf1 &    0 \\
   0 &    0 & \bf1 &    0 &    0 & \bf1 & \bf1 & \bf1 \\
\end{array}
\right)
=56;$