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Home  >>  CBSE XII  >>  Math  >>  Determinants
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Evaluate $ \begin{vmatrix} x & y & x+y\\ y & x+y & x\\ x+y & x & y \end{vmatrix}$

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Toolbox:
  • If each element of a row (or a column) of a determinant is multiplied by a constant k,then its value get multiplied by k.
  • By this property we can take out any common factor from any one row or any one column.
  • Elementary transformations can be done by
  • 1. Interchanging any two rows or columns. rows.
  • 2. Mutiplication of the elements of any row or column by a non-zero number
  • 3. The addition of any row or column , the corresponding elemnets of any other row or column multiplied by any non zero number.
Let $\Delta= \begin{vmatrix} x & y & x+y\\ y & x+y & x\\ x+y & x & y \end{vmatrix}$
Let us apply $R_1\rightarrow R_1+R_2+R_3$
$\Delta= \begin{vmatrix} 2x+2y & 2x+2y & 2x+2y\\ y & x+y & x\\ x+y & x & y \end{vmatrix}$
Taking 2(x+y) as the common factor ,we get
$\Delta=2(x+y) \begin{vmatrix} 1 & 1& 1\\ y & x+y & x\\ x+y & x & y \end{vmatrix}$
Apply $C_2\rightarrow C_2-C_1$ and $ C_2-C_3$
$\Delta=2(x+y) \begin{vmatrix} 1 & 0& 0\\ y & x & y\\ x+y & -y & x-y \end{vmatrix}$
Now expanding along $R_1$ we get,
$\Delta=1[x(x-y)+y^2]=(x^2-xy+y^2)2(x+y)$
$\;\;\;\;=2(x-y)(x^2-xy+y^2)$
$\;\;\;\;=2(x^3-x^2y+xy^2+x^2y-xy^2+y^3]$
$\;\;\;\;=2(x^3+y^3).$

 

answered Mar 1, 2013 by balaji.thirumalai
edited Mar 2, 2013 by vijayalakshmi_ramakrishnans
 

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