Question

Prove that the determinant `|(x,sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1, x)|` is independent of θ.

Answer 1

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.