Question

If the determinant of a matrix of order $3\times 3$ is formed by using the numbers 1 or -1 and minimum value of the determinant is $-\lambda$, then the value of $\lambda$ is ___ .

Answer 1

$Let\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|C_2\to C_2-\frac{a_{12}}{a_{11}}{C}_1;C_3\to C_3-\frac{a_{13}}{a_{11}}{C}_1\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& (a_{22}-\frac{a_{12}}{a_{11}}\times a_{21})& (a_{23}-\frac{a_{13}}{a_{11}}\times a_{21})\\ a_{31}& (a_{32}-\frac{a_{12}}{a_{11}}\times a_{31})& (a_{33}-\frac{a_{13}}{a_{11}}\times a_{31})\end{array}$

So minimum value = - 4.