1.

Prove that `|(1,a,a^2),(1,b,b^2),(1,c,c^2)|=(a-b)(b-c)(c-a)`

Answer» Let the given determinant be `Delta`. Then,
`Delta =|{:(1, a, a^(2)),(1, b, b^(2)),(1, c, c^(2)):}|`
`=|{:(1, a, a^(2)),(0, b-a, b^(2)-a^(2)),(0, c-a, c^(2)-a^(2)):}| ["applying" R_(2) to (R_(2)-R_(1))"and" R_(3) to (R_(3) -R_(1))]`
` = (b-a)(c-a) * |{:(1, a, a^(2)),(0, 1, b+a),(0, 1, c+a):}|`
`["taking (b-a) common from "R_(2) "and (c-a) common from "R_(3)]`
`=(b-a)(c-a) xx 1 * |{:(1, b+a), (1, c+a):}| " "["expanded by"C_(1)]`
`=(b-a)(c-a){(c+a)-(b+a)}`
`=(b-a)(c-a)(c-b) = (a-b)(b-c)(c-a)`
Hence, `Delta = (a-b)(b-c)(c-a)`


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