If a determinant of order 3 x 3 is formed by using the numbers 1 or -1, then the minimum value of the determinant is

Let $\mathrm{\Delta }=\left|\begin{array}{l}{\mathrm{a}}_{11}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_{12}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_{22}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_{32}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_{33}\end{array}\right|$

Applying ${\mathrm{C}}_2\to {\mathrm{C}}_2-\frac{{\mathrm{a}}_{12}}{{\mathrm{a}}_{11}}{\mathrm{C}}_1,{\mathrm{C}}_3\to {\mathrm{C}}_3-\frac{{\mathrm{a}}_{13}}{{\mathrm{a}}_{11}}{\mathrm{C}}_1$, we get

$\mathrm{\Delta }=\left|\begin{array}{lcc}{\mathrm{a}}_{11}& 0& 0\\ {\mathrm{a}}_{21}& \left({\mathrm{a}}_{22}-\frac{{\mathrm{a}}_{12}}{{\mathrm{a}}_{11}}\times {\mathrm{a}}_{21}\right)& \left({\mathrm{a}}_{23}-\frac{{\mathrm{a}}_{13}}{{\mathrm{a}}_{11}}{\mathrm{a}}_{21}\right)\\ {\mathrm{a}}_{31}& \left({\mathrm{a}}_{32}-\frac{{\mathrm{a}}_{12}}{{\mathrm{a}}_{11}}\times {\mathrm{a}}_{31}\right)& \left({\mathrm{a}}_{32}-\frac{{\mathrm{a}}_{13}}{{\mathrm{a}}_{11}}\times {\mathrm{a}}_{31}\right)\end{array}\right|$

which has minimum value of -4.