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प्रश्न
Prove that the determinant `|(x,sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1, x)|` is independent of θ.
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उत्तर
Let, Δ = `|(x,sintheta,costheta),(-sintheta,-x,1),(costheta,1,x)|`
= x(−x2 − 1) − sin θ(−x sin θ − cos θ) + cos θ(−sin θ + x cos θ)
= −x(x2 + 1) + x sin2 θ + sin θ cos θ − sin θ cos θ + x cos2 θ
= −x(x2 + 1) + x(sin2 θ + cos2 θ)
= −x(x2 + 1) + x
= −x[x2 + 1 − 1]
= −x3
Hence, the determinant is independent of θ.
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