First slide

Introduction to Determinants

Question

The system of the equations
                     $x - y \cos θ + z \cos 2 θ = 0$

                     $- x \cos θ + y - z \cos θ = 0$

                      $x \cos 2 θ - y \cos θ + z = 0$

Has non trivial solution for $θ$ equals

Easy

A

$π / 3$
B

$π / 6$
C

$2 π / 3$
D

$π / 12$

Solution

$| \begin{matrix} 1 & - \cos θ & \cos 2 θ \\ - \cos θ & 1 & - \cos θ \\ \cos 2 θ & - \cos θ & 1 \end{matrix} | = 0$
$A p p l y i n g C_{1} \to C_{1} - C_{3}, t h e n$
$\Rightarrow | \begin{matrix} 2 \sin^{2} θ & - \cos θ & \cos 2 θ \\ 0 & 1 & - \cos θ \\ - 2 \sin^{2} θ & - \cos θ & 1 \end{matrix} | = 0$
$\Rightarrow 2 \sin^{2} θ | \begin{matrix} 1 & - \cos θ & \cos 2 θ \\ 0 & 1 & - \cos θ \\ - 1 & - \cos θ & 1 \end{matrix} | = 0$
$\Rightarrow 2 \sin^{2} θ {1 (1 - \cos^{2} θ) - 1 (\cos^{2} θ - \cos 2 θ)} = 0$
$\Rightarrow 2 \sin^{2} θ {- 2 \cos^{2} θ + 1 + \cos 2 θ} = 0$
$\Rightarrow 2 \sin^{2} 0 {0} = 0 \Rightarrow 0 = 0$
Which is independent of $θ$ .

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