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

My question is simply the title:

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

• Since the determinant is the sum of $\,n!\,$ products of elements of the matrix chosen in a particular way, I'd say the maximum value is $\,n!\,$ ...but I've no example to offer. – DonAntonio Jun 20 '13 at 13:09
• $n!$ is impossible. – Mher Jun 20 '13 at 13:13
• There can't be any example for that – Mher Jun 20 '13 at 13:13
• An upper bound that is smaller than $n!$ is $n^{n/2}$. Google Hadamard's inequality. – KCd Jun 20 '13 at 13:14
• I can offer an example for $n-1.$ Taking $a_{ii}=0$ and $a_{ij}=1$ if $i\neq j.$ – Mher Jun 20 '13 at 13:16

Quoting my question in another thread:

In fact, I don't even know how large the determinant of a 0-1 matrix can be. The Hadamard's bound for the absolute determinant of an $n\times n$ 0-1 matrix is $\frac{(n+1)^{(n+1)/2}}{2^n}$ (online ref. 1 and ref. 2), and the bound is sharp if and only if there exists a Hadamard matrix of order $n+1$. Yet, to my knowledge, there is no known sharp upper bound for the absolute determinant of a general $n\times n$ 0-1 matrix.

• Hm, can someone explain (or direct me to someplace) to understand why this answer is a community wiki? I've not dealt with community wikis before, so just curious. – Calvin Lin Jun 20 '13 at 14:44
• @CalvinLin See an explanation here. In the present case I turned my answer into a community wiki one on purpose, because I didn't think simply quoting my own question verbatim deserves any reputation points. – user1551 Jun 20 '13 at 15:13

A few examples of $\{0,1\}$-matrices (with the largest determinants $-$ according to OEIS-A003432):

$n=2$: $\quad\det\left( \begin{array}{cc} \bf{1} & 0 \\ \bf{1} & \bf{1} \\ \end{array} \right) =1;$

$n=3$: $\quad\det\left( \begin{array}{ccc} \bf{1} & 0 & \bf{1} \\ \bf{1} & \bf{1} & 0 \\ 0 & \bf{1} & \bf{1} \\ \end{array} \right) =2;$

$n=4$: $\quad\det\left( \begin{array}{cccc} \bf{1} & 0 & \bf{1} & 0 \\ \bf{1} & \bf{1} & 0 & \bf{1} \\ 0 & \bf{1} & \bf{1} & 0 \\ 0 & 0& \bf{1} & \bf{1} \\ \end{array} \right) =3;$

$n=5$: $\quad\det\left( \begin{array}{ccccc} \bf{1} & 0 & \bf{1} & 0 & 0\\ \bf{1} & \bf{1} & 0 & \bf{1} & 0 \\ 0 & \bf{1} & \bf{1} & 0 &\bf{1}\\ 0 & 0 & \bf{1} & \bf{1} & 0 \\ \bf{1} & 0 & 0 & \bf{1} & \bf{1} \\ \end{array} \right) =5;$

$n=6$: $\quad\det\left( \begin{array}{ccccc} \bf{1} & 0 & \bf{1} & 0 & 0 & 0\\ \bf{1} & \bf{1} & 0 & \bf{1} & 0 & 0 \\ 0 & \bf{1} & \bf{1} & 0 &\bf{1} & 0\\ 0 & 0 & \bf{1} & \bf{1} & 0 & \bf{1}\\ \bf{1} & 0 & 0 & \bf{1} & \bf{1} & 0\\ \bf{1} & \bf{1} & 0 & 0 & \bf{1} & \bf{1} \\ \end{array} \right) =9;$

$n=7$: $\quad\det\left( \begin{array}{ccccc} \bf1 & 0 & \bf1 & 0 & 0 & \bf1 & \bf1 \\ \bf1 & \bf1 & 0 & \bf1 & 0 & 0 & \bf1 \\ \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 & 0 \\ 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 \\ 0 & 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 \\ \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 & 0 \\ 0 & \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 \\ \end{array} \right) =32;$

$n=8$: $\quad\det\left( \begin{array} \bf1 & 0 & \bf1 & 0 & 0 & \bf1 & \bf1 & 0 \\ \bf1 & \bf1 & 0 & \bf1 & 0 & 0 & \bf1 & \bf1 \\ \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 & 0 & \bf1 \\ 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 & 0 \\ 0 & 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 & 0 \\ \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 & 0 & \bf1 \\ 0 & \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 & 0 \\ 0 & 0 & \bf1 & 0 & 0 & \bf1 & \bf1 & \bf1 \\ \end{array} \right) =56;$

• Very nice! I was also looking for patterns of symmetry but didn't search too long... (I stopped after having brute-forced $n=4$ with $2^{16}$ variants of the matrix ... ;-)) – Gottfried Helms Jun 21 '13 at 9:47
• Yes, we can find first 8 matrices among striped matrices ($\searrow \searrow \searrow$). What about others $n$ ??? – Oleg567 Jun 21 '13 at 9:51