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[[b+c,a-b,a],[c+a,b-c,b],[a+b,c-a,c]] = ...

`[[b+c,a-b,a],[c+a,b-c,b],[a+b,c-a,c]] = 3abc - a^3 - b^3 - c^3`

Text Solution

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`L.H.S. = |[b+c,a-b,a],[c+a,b-c,b],[a+b,c-a,c]|`
Applying `C_2->C_2-C_3`
`= |[b+c,-b,a],[c+a,-c,b],[a+b,-a,c]|`
Applying `C_1->C_+C_2`
`= |[c,-b,a],[a,-c,b],[b,-a,c]|`
`=-1 |[c,b,a],[a,c,b],[b,a,c]|`
`= - 1[c^3-abc - bac+b^3 + a^3 - abc]`
`= 3abc - a^3-b^3-c^3 = R.H.S.`
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