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

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

Text Solution

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`|(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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