Question

Prove that: $\left|\begin{array}{ccc}1& x& y+z\\ 1& y& z+x\\ 1& z& x+y\end{array}\right|=0$

Answer 1

To prove: $\left|\begin{array}{ccc}1& x& y+z\\ 1& y& z+x\\ 1& z& x+y\end{array}\right|=0$

Lets solve $\left|\begin{array}{ccc}1& x& y+z\\ 1& y& z+x\\ 1& z& x+y\end{array}\right|$ as

$=1\left[y\right(x+y)-z(z+x\left)\right]-x[x+y-(z+x\left)\right]+(y+z)[z-y]$

$=[xy+y^2-z^2-zx]-[x^2+xy-xz-x^2]+[yz-y^2+z^2-zy]$

$=xy+y^2-z^2-zx-xy+xz+yz-y^2+z^2-zy$

$=0$

Therefore, $\left|\begin{array}{ccc}1& x& y+z\\ 1& y& z+x\\ 1& z& x+y\end{array}\right|=0$

Hence, proved.