1.

Prove that the determinant `Delta =|{:(x,,sintheta,,cos theta),(-sin theta,,-x,,1),(cos theta,,1,,x):}|` is independent of `theta`.

Answer» Expanding along the first row we get
`Delta =|{:(x,,sin0,,cos 0),(-sin 0,,-x,,1),(cos 0,,1,,x):}|`
`=x(-x^(2)-1) -sin 0( -x sin 0- cos 0)`
` + cos 0 (-sin 0 +x cos 0)`
`=-x^(3) +x + x sin^(2)0 + sin 0 cos 0 -sin 0 cos + x cos^(2) 0`
`= -x^(3) + x+ x(sin^(2) 0+ cos^(2)0 )`
`= -x^(3) + 2x`
Hence `Delta ` is independent of 0.


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