Prove the identities: `|[a, b, c],[ a-b,b-c,c-a],[ b+c,c+a, a+b]|=a^3+b^3+c^3-3a b c`

Explanation

`|(a,b,c),(a-b,b-c,c-a),(b+c, c+a,a+b)|`

`c_1-> c_1 + c_2 + c_3`

`|(a+b+c, b,c),(0,b-c,c-a),(2(a+b+c), c+a, a+b)|`

`= (a+b+c)|(1,b,c),(0,b-c, c-a),(2,c+a,a+b)|`

`r_3 -> r_3 - 2r_1`

`(a+b+c)|(1,b,c),(0, b-c, c-a),(0, c+a-2b, a+b-2c)|`

`= (a+b+c)[(b-c)(a+b-2c)- (c-a)(c+a-2b)]`

`= (a+b+c)[ab+b^2 - 2ac - ac - bc +2c^2 - c^2 - ac+ 2bc+ ac + a^2 - 2ab]`

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