First slide

Introduction to Determinants

Question

Suppose $a, b, c$ are sides of a scalene triangle. Let $Δ = |\begin{array}{lll} a b c \\ b c a \\ c a b \end{array}|$ .Then

Moderate

A

$Δ \leq 0$
B

$Δ < 0$
C

$Δ > 0$
D

$Δ \geq 0$

Solution

Using $C_{1} \to C_{1} + C_{2} + C_{3}$ we get
$Δ = (a + b + c) Δ_{1}$
where
$Δ_{1} = |\begin{array}{lll} 1 b c \\ 1 c a \\ 1 a b \end{array}|$
Applying $R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - R_{1}$ we get
$\begin{array}{l} Δ_{1} = |\begin{array}{ccc} 1 & b & c \\ 0 & c - b & a - c \\ 0 & a - b & b - c \end{array}| \\ = - (b - c)^{2} - (a - b) (a - c) \\ = - (a^{2} + b^{2} + c^{2} - b c - c a - a b) \end{array}$
$\Rightarrow Δ_{1} = - \frac{1}{2} [(b - c)^{2} + (c - a)^{2} + (a - b)^{2}] < 0$
$As a + b + c > 0$ , we get
$Δ = (a + b + c) Δ_{1} < 0$

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