The determinant | x sin cos - sin - x 1 cos 1 x | is:

Question

The determinant $| \begin{matrix} x & \sin θ & \cos θ \\ - \sin θ & - x & 1 \\ \cos θ & 1 & x \end{matrix} |$ is:

Smart Achievers · Accepted Answer

(1)

Ans. Independent of $θ$ only

Explanation:

Let $Δ =$ $| \begin{matrix} x & \sin θ & \cos θ \\ - \sin θ & - x & 1 \\ \cos θ & 1 & x \end{matrix} |$

$= x (- x^{2} - 1) - \sin θ (- x \sin θ - \cos θ) + \cos θ (- \sin θ + x \cos θ)$

$= - x^{3} - x + x \sin^{2} θ + \sin θ \cos θ - \sin θ \cos θ + x \cos^{2} θ$

$= - x^{3} - x + x (\sin^{2} θ + \cos^{2} θ)$

$= - x^{3} - x + x$ $[∵ \sin^{2} θ + \cos^{2} θ = 1]$

$= - x^{3},$ which is independent of $θ$ .

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