If a, b, c are all positive and not all equal then the value of the determinant is

Engineering

Mathematics

Properties of Determinant

Inequalities

Question

If a, b, c are all positive and not all equal then the value of the determinant $[\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}]$ is

> 0

< 0

cannot be determined

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Let

| A | = [\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}]

C₁ → C₁ + C₂ + C₃

$∣ A ∣ = ∣ \begin{array}{ccc} a + b + c & b & c \\ a + b + c & c & a \\ a + b + c & a & b \end{array} ∣$
$= (a + b + c) ∣ \begin{array}{ccc} 1 & b & c \\ 1 & c & a \\ 1 & a & b \end{array} ∣$

R₁ → R₁ – R₂ R₂ → R₂ – R₃

= (a + b + c) ∣ \begin{array}{ccc} 0 & b - a & c - a \\ 0 & c - a & a - b \\ 1 & a & b \end{array} ∣

= (a + b + c)[(b – a)(a – b) – (c – a)²]
∴ |A| = – (a + b + c)[(a – b)² + (c – a)²]
Now, (a – b)², (c – a)² > 0 and a + b + c > 0 as a, b, c > 0

∴ |A| < 0 0

Hence,

[\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}] < 0