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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS4 Marks
Question
Using the property of determinants and without expanding, prove that:
$\begin{vmatrix}b+c&q+r&y+z\\c+a&r+p&z+x\\a+b&p+q&x+y\end{vmatrix}=2\begin{vmatrix}a&p&x\\b&q&y\\c&r&z\end{vmatrix}$
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Answer

$\text{L.H.S.}= \begin{vmatrix}(b+c)&q+r&y+z\$c+a)&r+p&z+x\$a+b)&p+q&x+y\end{vmatrix}\ \text{operating}\ \text{R}_1\rightarrow \text{R}_1+\text{R}_2+\text{R}_3$
$=\begin{vmatrix}b+c+c+a+a+b&q+r+r+p+p+q&y+z+z+x+x+y\\c+a&r+p&z+x\\a+b&p+q&x+y\end{vmatrix}$
$=\begin{vmatrix}2(a+b+c)&2(p+q+r)&2(x+y+z)\\c+a&r+p&z+x\\a+b&p+q&x+y\end{vmatrix}$
$=2\begin{vmatrix}(a+b+c)&(p+q+r)&(x+y+z)\\c+a&r+p&z+x\\a+b&p+q&x+y\end{vmatrix}$
$=2\begin{vmatrix}b&q&y\\c+a&r+p&z+x\\a+b&p+q&x+y\end{vmatrix}\ \left[\text{operating}\ \text{R}_1\rightarrow \text{R}_1-\text{R}_2\right]$
$=2\begin{vmatrix}b&q&y\\c+a&r+p&z+x\\a&p&x\end{vmatrix}\ \left[\text{operating}\ \text{R}_3\rightarrow \text{R}_3-\text{R}_1\right]$
$=2\begin{vmatrix}b&q&y\\c&r&z\\a&p&x\end{vmatrix}\ \left[\text{operating}\ \text{R}_2\rightarrow \text{R}_2-\text{R}_3\right]$
$=-2\begin{vmatrix}b&q&y\\a&p&x\\c&r&z\end{vmatrix}\ \left[\text{interchanging}\ \text{R}_2\ \text{and} \ \text{R}_3\right]$
$=-(-2)\begin{vmatrix}a&p&x\\b&q&y\\c&r&z\end{vmatrix}\ \left[\text{interchanging}\ \text{R}_1\ \text{and} \ \text{R}_2\right]$
$=2\begin{vmatrix}a&p&x\\b&q&y\\c&r&z\end{vmatrix}=\text{R.H.S.}$

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