Question

Show that $\left|\begin{array}{ccc}y+z& x& y\\ z+x& z& x\\ x+y& y& z\end{array}\right|=\left(x+y+z\right){\left(x-z\right)}^2$

Answer 1

$$\begin{gathered} \mathrm{Let}∆=\left|y+zxy \\ z+xzx \\ x+yyz\right| \\ \Rightarrow ∆=\left|2\left(x+y+z\right)x+y+zx+y+z \\ z+xzx \\ x+yyz\right|\left[\mathrm{Applying}{\mathrm{R}}_1\to {\mathrm{R}}_1+{\mathrm{R}}_2+{\mathrm{R}}_3\right] \\ =\left(x+y+z\right)\left|211 \\ z+xzx \\ x+yyz\right| \\ =\left(\mathrm{x}+y+\mathrm{z}\right)\left|011 \\ 0\mathrm{z}\mathrm{x} \\ x-z\mathrm{y}\mathrm{z}\right|\left[\mathrm{Applying}{C}_1\to C_1-C_2-C_3\right] \\ =\left(\mathrm{x}+y+\mathrm{z}\right)\left\{\left(x-z\right)\times \left|\begin{array}{cc}1& 1\\ z& x\end{array}\right|\right\}\left[\mathrm{Expanding}\mathrm{along}{C}_1\right] \\ =\left(\mathrm{x}+y+\mathrm{z}\right){\left(x-z\right)}^2 \end{gathered}$$