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Without expanding the determinant, Pro...

Without expanding the determinant,
Prove that `|[a,a^2, bc],[b,b^2,ca],[c,c^2,ab]|=` `|[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|`

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L.H.S.`=1/(abc)|[a^2,a^3,abc],[b^2,b^3,abc],[c^2,c^3,abc]|=1/(abc)(abc)|[a^2,a^3,1],[b^2,b^3,1],[c^2,c^3,1]|=|[a^2,a^3,1],[b^2,b^3,1],[c^2,c^3,1]|`
`=|[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|` (by `C_2 harrC_3`and then by `C_1harrC_2`)=R.H.S.
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