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Prove that |(a+b,b+c,c+a),(b+c,c+a,a+b),...

Prove that `|(a+b,b+c,c+a),(b+c,c+a,a+b),(c+a,a+b,b+c)|=- 2 (a^3+ b^3+c^3-3abc)`

Text Solution

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`L.H.S. = |[a+b,b+c,c+a],[b+c,c+a,a+b],[c+a,a+b,b+c]|`
`=|[a,b,c],[b,c,a],[c,a,b]|+|[b,c,a],[c,a,b],[a,b,c]|`
`=(-1)|[a,c,b],[b,a,c],[c,b,a]|+(-1)|[c,b,a],[a,c,b],[b,a,c]|`
`=(-1)^2|[c,a,b],[a,b,c],[b,c,a]|+(-1)^2|[c,a,b],[a,b,c],[b,c,a]|`
`=|[c,a,b],[a,b,c],[b,c,a]|+|[c,a,b],[a,b,c],[b,c,a]|`
`=2|[c,a,b],[a,b,c],[b,c,a]|`
`=2[c(ba-c^2)-a(a^2-bc)+b(ac-b^2)]`
`=2[3cab-c^3-a^3-b^3]`
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