Question

Prove that the determinant is independent of θ .

Answer 1

The given determinant is,

| x sinθ cosθ −sinθ −x 1 cosθ 1 x |

Simplifying the above determinant,

Δ=| x sinθ cosθ −sinθ −x 1 cosθ 1 x | =x( − x 2 −1 )−sinθ( −xsinθ−cosθ )+cosθ( −sinθ+xcosθ ) =− x 3 −x+x sin 2 θ+sinθ⋅cosθ−sinθ⋅cosθ+x cos 2 θ =− x 3 −x+x( sin 2 θ+ cos 2 θ )

Since sin 2 θ+ cos 2 θ=1, thus,

Δ=− x 3 −x+x⋅1 =− x 3

Hence, Δ is independent of θ.