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Q : 9        Evaluate  \begin{vmatrix} x & y &x+y \\ y & x+y &x \\ x+y & x & y \end{vmatrix}

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We have determinant  \triangle = \begin{vmatrix} x & y &x+y \\ y & x+y &x \\ x+y & x & y \end{vmatrix}

Applying row transformations;  R_{1} \rightarrow R_{1}+R_{2}+R_{3} , we have then;

\triangle = \begin{vmatrix} 2(x+y) & 2(x+y) &2(x+y) \\ y & x+y &x \\ x+y & x & y \end{vmatrix}

Taking out the common factor 2(x+y) from the row first.

= 2(x+y)\begin{vmatrix} 1 & 1 &1 \\ y & x+y &x \\ x+y & x & y \end{vmatrix}

Now, applying the column transformation; C_{1} \rightarrow C_{1} - C_{2} and  C_{2} \rightarrow C_{2} - C_{1} we have ;

= 2(x+y)\begin{vmatrix} 0 & 0 &1 \\ -x & y &x \\ y & x-y & y \end{vmatrix}

Expanding the remaining determinant;

= 2(x+y)(-x(x-y)-y^2) = 2(x+y)[-x^2+xy-y^2]

= -2(x+y)[x^2-xy+y^2] = -2(x^3+y^3).

 

 

 

 

Posted by

Divya Prakash Singh

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