First slide

Introduction to Determinants

Question

$Let α, β and γ be the roots of the equation x^{3} - x^{2} + 3 x + 1 = 0 . Then the value of |\begin{array}{lll} γ β α + 2 β + 2 γ \\ α γ β + 2 α + 2 γ \\ β α γ + 2 α + 2 β \end{array}| is equal to$

Moderate

A

-10
B

-8
C

7
D

9

Solution

$\begin{array}{l} Δ = |\begin{array}{lll} γ & β & α + 2 β + 2 γ \\ α & γ & β + 2 α + 2 γ \\ β & α & γ + 2 α + 2 β \end{array}| \\ Applying C_{3} \to C_{3} - 2 C_{1} - 2 C_{2}, we get \\ Δ = |\begin{array}{lll} γ & β & α \\ α & γ & β \\ β & α & γ \end{array}| \\ = (α + β + γ) (α^{2} + β^{2} + γ^{2} - α β - β γ - γ α) \\ = (α + β + γ) [(α + β + γ)^{2} - 3 (α β + β γ + γ α)] \\ = 1 \times (1^{2} - 3 \times (3)) \\ = - 8 \end{array}$

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