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By using properties of determinants. Sho...

By using properties of determinants. Show that:
(i) `|[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|=(a-b)(b-c)(c-a)`
(ii) `|[1, 1, 1],[a, b, c],[ a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a+b+c)`

Text Solution

Verified by Experts

(i) `L.H.S.= |[1,a,a^2],[1,b,b^2],[1,c,c^2]|`
Applying `R_1->R_1-R_2` and `R_2->R_2-R_3`
`= |[0,a-b,a^2-b^2],[0,b-c,b^2 - c^2],[1,c,c^2]|`
`= |[0,a-b,(a-b)(a+b)],[0,b-c,(b-c)(b+c)],[1,c,c^2]|`
`= (a-b)(b-c)|[0,1,(a+b)],[0,1,(b+c)],[1,c,c^2]|`
`= (a-b)(b-c)[b+c-a-b]`
`= (a-b)(b-c)(c-a) = R.H.S.`

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