Q. 133.8( 19 Votes )

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Answer :

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, C1C1+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, R3R3-2R1


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


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