First slide

Introduction to Determinants

Question

If determinant $|\begin{array}{ccc} \cos (θ + ϕ) & - \sin (θ + ϕ) & \cos 2 ϕ \\ \sin θ & \cos θ & \sin ϕ \\ - \cos θ & \sin θ & \cos ϕ \end{array}|$ is

Moderate

A

non-negative
B

independent of $θ$
C

independent of $ϕ$
D

none of these

Solution

Applying $R_{1} \to R_{1} + \sin ϕ (R_{2}) + \cos ϕ (R_{3})$ ,
$\begin{matrix} f (x) = Δ = |\begin{array}{ccc} 0 & 0 & \cos 2 ϕ + 1 \\ \sin θ & \cos θ & \sin ϕ \\ - \cos ϕ & \sin θ & \cos ϕ \end{array}| \\ = (\cos 2 ϕ + 1) (\sin^{2} θ + \cos^{2} θ) \\ = (1 + \cos 2 ϕ) \end{matrix}$
Hence, $∆$ is independent of $θ$ .

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