First slide

Introduction to Determinants

Question

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

Moderate

A

-2
B

-4
C

0
D

-8

Solution

Let $Δ = |\begin{array}{lll} a_{11} a_{12} a_{13} \\ a_{21} a_{22} a_{23} \\ a_{31} a_{32} a_{33} \end{array}|$
Applying $C_{2} \to C_{2} - \frac{a_{12}}{a_{11}} C_{1}, C_{3} \to C_{3} - \frac{a_{13}}{a_{11}} C_{1}$ , we get
$Δ = |\begin{array}{lcc} a_{11} & 0 & 0 \\ a_{21} & (a_{22} - \frac{a_{12}}{a_{11}} \times a_{21}) & (a_{23} - \frac{a_{13}}{a_{11}} a_{21}) \\ a_{31} & (a_{32} - \frac{a_{12}}{a_{11}} \times a_{31}) & (a_{32} - \frac{a_{13}}{a_{11}} \times a_{31}) \end{array}|$
which has minimum value of -4.

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