The largest value of a third order determinant whose elements are equal to `1 or 0` is

Explanation

Let
` Delta=|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}|` be a determinant of order 3. Then
`Delta =(a_(1) b_(2) c_(3) +a_(3)b_(1)c_(2) + a_(2)b_(3)c_(1))-(a_(1)b_(3)c_(2)+a_(2)b_(1)c_(3)+a_(3)b_(2)c_(1))`
since each element of `Delta` is maximum when the value of each term in the first bracket is 1 and the value of each term in the second bracket is zero. But `a_(1)b_(2)c_(3) = a_(3)b_(1)c_(1)= 1` implies that every element of the determinant `Delta` is 1 and in that case `Delta =0` Thus we may have
` Delta=|{:(0,,1,,1),(1 ,,0,,1),(1,,1,,0):}|`