Question

Find the maximum value of $3\times 3$ determinant whose elements are $0,1$.

Answer 1

Identify the determinants and its maximum value

Case I: If all the entries in one row are $0$, then the value of the determinants is $0$.

For example $D=\left|\begin{array}{ccc}0& 0& 0\\ 1& 0& 1\\ 1& 1& 0\end{array}\right|=0$
Case II: If all the entries in one row are $1$ and non of the rows having all entries $0$, then the maximum value of determinant is $1$.

Case III: If exactly one $1$ is in each one row, then the maximum value is $1$.
Case IV: If exactly two $1$ is in one row, then the maximum value is $2$.

For example $D=\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right|=0\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|-1\left|\begin{array}{cc}1& 1\\ 1& 0\end{array}\right|+1\left|\begin{array}{cc}1& 0\\ 1& 1\end{array}\right|=0+1+1=2$

Hence, the largest value of the determinant with entries either $0$ or $1$ is $2$.