Question

Prove the following identities:
$\left|\begin{array}{ccc}b+c& c+a& a+b\\ q+r& r+p& p+q\\ y+z& z+x& x+y\end{array}\right|=2\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{ccc}b+c& c+a& a+b\\ q+r& r+p& p+q\\ y+z& z+x& x+y\end{array}\right|$

$\Rightarrow \Delta =\left|\begin{array}{ccc}b& c& a\\ q& r& p\\ y& z& x\end{array}\right|+\left|\begin{array}{ccc}c& a& b\\ r& p& q\\ z& x& y\end{array}\right|$

$\Rightarrow \Delta ={\Delta }_1+{\Delta }_2$

${\Delta }_1=\left|\begin{array}{ccc}b& c& a\\ q& r& p\\ y& z& x\end{array}\right|$

Interchanging $C_1$ and $C_3$

$\Rightarrow {\Delta }_1=-\left|\begin{array}{ccc}a& c& b\\ p& r& q\\ x& z& y\end{array}\right|$

Interchanging $C_2$ and $C_3$

$\Rightarrow {\Delta }_1=\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$

similarly in ${\Delta }_2$ interchange $C_2$ and $C_1$ and then $C_2$ and $C_3$

$\Rightarrow {\Delta }_2=\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$

$\Delta ={\Delta }_1+{\Delta }_2$

$\Rightarrow \Delta =2\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$ $\left[henceproved\right]$