Question

Prove that the determinant `[xsinthetacostheta-sintheta-x1costheta1x]`is independent of 0.

Answer 1

`[{:(x,"sin"theta,"cos"theta),("-sin"theta,-x,1),("cos"theta,1,x):}]`
`[{:(-x,1),(1,x):}]"-sin"theta[{:("-sin"theta,1),("cos"theta,x):}]"+cos"theta|{:("-sin"theta,-x),("cos"theta,1):}|`
`=x(-x^(2)-1)-"sin"theta(-x"sin"theta-"cos"theta)+"cos"theta(-"sin"theta"+x"cos"theta)`
`=-x^(3)-x+x"sin"^(2)theta+"sin"theta"cos"theta-"sin"theta"cos"theta+x"cos"^(2)theta`
`=-x^(3)-x+x(sin^(2)theta+cos^(2)theta)`
`=-x^(3)-x+x=-x^(3)`