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Multiplication of Matrices

If a, b, cϵ...

Question

If $a, b, c ϵ R$ and $a^{2} + b^{2} - a b - a - b + 1 \leq 0$ and $α + β + γ = 0$ , then
$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & cos γ & cos β cos γ & a & cos α cos α & cos β & b \end{matrix} ∣ ∣ ∣ ∣ ∣$ equals

A

ab

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B

1

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C

0

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D

-1

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Solution

The correct option is A 0
$a^{2} + b^{2} - a b - a - b + 1 \leq 0 \Rightarrow a^{2} + b^{2} - 2 a b + a b - a - b + 1 \leq 0 \Rightarrow a = b = 1$
As $α + β + γ = 0$
Let $α = B - C, β = C - A, γ = A - B$
$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & cos (A - B) & cos (C - A) cos (A - B) & 1 & cos (B - C) cos (B - C) & cos (C - A) & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ ∣ \begin{matrix} 1 & c o s A c o s B + s i n A s i n B & c o s C c o s A + s i n C s i n A c o s A c o s B + s i n A s i n B & 1 & c o s B c o s C + s i n B s i n C c o s B c o s C + s i n B s i n C & c o s C c o s A + s i n C s i n A & 1 \end{matrix} ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ ∣ \begin{matrix} c o s A & s i n A & 0 c o s B & s i n B & 0 c o s C & s i n C & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} c o s A & c o s B & c o s C s i n A & s i n B & s i n C 0 & 0 & 0 \end{matrix} ∣ ∣ ∣ ∣ = 0$

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