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Using properties of determinants, pro...

Using properties of determinants, prove the following: `|1xx^2x^2 1xxx^2 1|=(1-x^3)^2`

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`L.H.S.=|{:(1,x,x^(2)),(x^(2),1,x),(x,x^(2),1):}|=|{:(1+x+x^(2),x,x^(2)),(x^(2)+1+x,1,x),(x+x^(2)+1,x^(2),1):}|`
`(C_(1)toC_(1)+C_(2)+C_(3))`
`=(1+x+x^(2))|{:(1,x,x^(2)),(1,1,x),(1,x^(2),1):}|`
`=(1+x+x^(2))|{:(1,x,x^(2)),(0,1-x,x-x^(2)),(0,x^(2)-x,1-x^(2)):}|`
`(R_(2)toR_(2)-R_(1),R_(3)toR_(3)-R_(1))`
`=(1+x+x^(2))|{:(1,x,x^(2)),(0,1-x,x(1-x)),(0,-x(1-x),(1-x)(1+x)):}|`
`=(1+x+x^(2))(1-x)(1-x)|{:(1,x,x^(2)),(0,1,x),(0,-x,1+x):}|`
`=(1+x+x^(2))(1-x).1|{:(1,x),(-x,1+x):}|`
`=(1+x+x^(2))(1-x)^(2)(1+x+x^(2))`
`=[(1+x+x^(2))(1-x)^(2)]=(1-x^(3))^(2)`
R.H.S.
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