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Question

Prove that the determinant ∣ ∣xsin θcos θsin θx1cos θ1x∣ ∣ is independent of θ.


Solution

Let A=∣ ∣xsin θcos θsin θx1cos θ1x∣ ∣ 
Expanding to corresponding first row, we get
A=xx11xsin θsin θ1cos θx+cos θsin θxcos θ1
=x(x21)sin θ(x sin θcos θ)+cos θ(sin θ+x cos θ)=x3x+x sin2 θ+sin θcos θsin θcos θ+xcos2 θ=x3x+x(sin2 θ+cos2 θ)=x3x+x   (sin2 θ+cos2 θ=1)
=x3.  Hence, A is independent of θ.

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