Question

Prove the following identities:

$\left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|=4xyz$

Answer 1

$$\begin{gathered} \mathrm{LHS}: \\ \left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right| \\ =\left|\begin{array}{ccc}y+z-z-y& z-z-x-x& y-x-x-y\\ z& z+x& x\\ y& x& x+y\end{array}\right|\left[\mathrm{Applying}{R}_1\to R_1-R_2-R_3\right] \\ =\left|\begin{array}{ccc}0& -2x& -2x\\ z& z+x& x\\ y& x& x+y\end{array}\right| \\ =-2x\left|\begin{array}{ccc}0& 1& 1\\ z& z+x& x\\ y& x& x+y\end{array}\right|\left[\mathrm{Taking}-2x\mathrm{common}\mathrm{from}{R}_1\right] \\ =-2x\left|\begin{array}{ccc}0& 0& 1\\ z& z& x\\ y& -y& x+y\end{array}\right|\left[\mathrm{Applying}{C}_2\to C_2-C_3\right] \\ =-2x\left(-zy-zy\right)\left[\mathrm{Expanding}\mathrm{along}\mathrm{first}\mathrm{row}\right] \\ =4xyz \\ =\mathrm{RHS} \\ \therefore \left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|=4xyz \end{gathered}$$