1.

Prove that`|a b+c a^2b c+a b^2c a+b c^2|=-(a+b+c)xx(a-b)(b-c)(c-a)dot`

Answer» We have `Delta = |{:(a,,b+c,,a^(2)),(b,,c+a,,b^(2)),(c,,a+b,,c^(2)):}|`
Applying `C_(2) to C_(2) +C_(1)` and then taking `(a+b+c)` common from `C_(2)` we get
` Delta =(a+b+c)xx |{:(a,,1,,a^(2)),(b,,1,,b^(2)),(c,,1,,c^(2)):}|`
`=- (a+b+c) xx |{:(1,,a,,a^(2)),(1,,b,,b^(2)),(1,,c,,c^(2)):}| " (Applying " C_(1) to C_(2)")"`
`=- (a+b+c)(a-b)(b-c)(c-a)`


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