First slide

Introduction to Determinants

Question

$Let Δ = \det (\begin{array}{lll} a b c \\ b c a \\ c a b \end{array}) . Then which of the following statements is false for a, b, c \in R ?$

Moderate

A

$Δ > 0 i f a = - 1, b = - 2, c = - 3$
B

$Δ < 0 i f a = 1, b = 2, c = 3$
C

$Δ = 0 if and only if a = b = c$
D

$Δ = 0 if and only if a^{3} + b^{3} + c^{3} = 3 a b c$

Solution

$\begin{array}{l} Δ = - (a^{3} + b^{3} + c^{3} - 3 a b c) \\ = \frac{- 1}{2} (a + b + c) [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}] \end{array}$

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