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Consider

Using the property that if the equalities of corresponding elements of other rows (or columns) are added to every element of any row (or column) of a determinant, then the value of determinant remains the same

Using column transformation, C₁→C₁+C₂+C₃

we get,

Using the property that if each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

Taking out factor (a + b + c) from C₁,

we get,

Using row transformation, R₃→R₃-2R₁

we get,

Expanding along C₁, we get

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

= (a + b + c)[ab+b²-2bc-ac-bc+2c²-c²-ac+2bc+ac+a²-2ab]

= (a + b + c)[a²+b²+c²-ab-bc-ac]

= a³+b³+c³-3abc