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Prove that the determinant {vmatrix} x & & - & -x & 1

Question

Prove that the determinant $∣ ∣ x - sin θ cos θ sin θ - x 1 cos θ 1 x ∣ ∣$ is independent of $θ$ .

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Prove that the determinant

∣ ∣ x - sin θ cos θ sin θ - x 1 cos θ 1 x ∣ ∣

is independent of

θ

14 mins ago

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14 mins ago

Text solutionVerified

∣ ∣ x - sin θ cos θ sin θ - x 1 cos θ 1 x ∣ ∣

Expanding along the first row, we get

Δ = 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)

\Rightarrow Δ = - x^{3}

(

∵ sin^{2} θ + cos^{2} θ = 1

)

Hence, the given determinant is independent of

θ

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Question Text	Prove that the determinant $∣ ∣ x - sin θ cos θ sin θ - x 1 cos θ 1 x ∣ ∣$ is independent of $θ$ .
Answer Type	Text solution:1
Upvotes	150