If abc ≠ 0 and if then

Engineering

Mathematics

Introduction to Determinants

Properties of Determinant

Exponential and Basic Maths questions

Question

If abc ≠ 0 and if $∣ \begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array} ∣ = 0$ then $\frac{a^{3} + b^{3} + c^{3}}{abc}$

– 3

– 2

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$∣ \begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array} ∣$
= a(bc – a²) – b(b² – ac) + c(ab – c²)
= 3abc – a³ – b³ – c³
Given,
$∣ \begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array} ∣ = 0$
So, $\frac{a^{3} + b^{3} + c^{3}}{abc} = 3$