First slide

Introduction to Determinants

Question

If $f (x) = a + bx + {cx}^{2}$ and $α, β, γ$ are the roots of the equation $x^{3} = 1, then |\begin{array}{lll} a b c \\ b c a \\ c a b \end{array}|$ is equal to

Moderate

A

$f (α) + f (β) + f (γ)$
B

$f (α) f (β) + f (β) f (γ) + f (γ) f (α)$
C

$f (α) f (β) f (γ)$
D

$- f (α) f (β) f (γ)$

Solution

$\begin{matrix} |\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}| = - (a^{3} + b^{3} + c^{3} - 3 abc) \\ = - (a + b + c) (a + {bω}^{2} + cω) (a + bω + {cω}^{2}) \end{matrix}$
(where $ω$ is cube roots of unity)
$= - f (α) f (β) f (γ) [∵ α = 1, β = ω, γ = ω^{2}]$

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