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Home  >>  CBSE XII  >>  Math  >>  Model Papers
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Using the properties of determinants, prove the following : \[ \begin{vmatrix} b+c & c+a & a+b \\ c+a & a+b & c+a \\ a+b & b+c & c+a \end{vmatrix} = 2(a+b+c)(ab+bc+ca-a^2-b^2-c^2).\]

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Toolbox:
  • Elementary transformation can be done by interchanging any two rows or any two columns.
  • The addition to the elements of any row or column the corresponding elements of any other row or column multiplied by any non-zero number.
Given:
 
$\Delta=\begin{vmatrix}b+c & c+a & a+b\\c+a & a+b & b+c\\a+b & b+c & c+a\end{vmatrix}$
 
Apply $R_1\rightarrow R_1+R_2+R_3$
 
$\Delta=\begin{vmatrix}2a+2b+2c & 2a+2b+2c & 2a+2b+2c\\c+a & a+b & b+c\\a+b & b+c & c+a\end{vmatrix}$
 
Taking 2(a+b+c) as a common factor from $R_1$
 
$\Delta=2(a+b+c)\begin{vmatrix}1 & 1 & 1\\c+a & a+b & b+c\\a+b & b+c & c+a\end{vmatrix}$
 
Apply $C_1\rightarrow C_1-C_2$ and $C_2\rightarrow C_2-C_3$
 
$\Delta=2(a+b+c)\begin{vmatrix}0 & 0 & 1\\c-b & a-c & b+c\\a-c & b-a & c+a\end{vmatrix}$
 
Now expanding along $R_1$
 
$\;\;\;=2(a+b+c)[0-0+1[(c-b)(b-a)-(a-c)(a-c)]$
 
$\;\;\;=2(a+b+c)[bc-ac-b^2+ab-a^2+ac+ac-c^2]$
 
$\;\;\;=2(a+b+c)[ab+bc+ac-a^2-b^2-c^2]$
 
Hence proved.

 

answered Mar 11, 2013 by sreemathi.v
 

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