Question

Find the largest value of a third-order determinant whose element are $0$ or $1$.

Answer 1

$\Delta =\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

$=(a_1{b}_2{c}_3+a_3{b}_1{c}_2+a_2{b}_3{c}_1)-(a_1{b}_3{c}_2+a_2{b}_1{c}_3+a_3{b}_2{c}_1)$

$\Delta =\left|\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right|$

$=\left[\right(0\left)\right(0\left)\right(0)+(1\left)\right(1\left)\right(1)+(1\left)\right(1\left)\right(1\left)\right]-\left[\right(0\left)\right(1\left)\right(1)+(1\left)\right(1\left)\right(0)+(1\left)\right(0\left)\right(1\left)\right]$

$=[0+1+1]-[0+0+0]$

$=2$

$\therefore$ The largest value of a third-order determinant whose element are $0$ or $1$ is $2$