Question

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

Answer 1

We have to prove,
$\left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|=4xyz$
$\therefore$ LHS $=\left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|$
$=\left|\begin{array}{ccc}y+z+z+y& z& y\\ z+z+x+x& z+x& x\\ y+x+x+y& x& x+y\end{array}\right|$ $[\because C_1\to C_1+C_2+C_3]$
$=2\left|\begin{array}{ccc}(y+z)& z& y\\ (z+x)& z+x& x\\ (x+y)& x& x+y\end{array}\right|$ [taking 2 common from $C_1$]
$=2\left|\begin{array}{ccc}y& z& y\\ 0& z+x& x\\ y& x& x+y\end{array}\right|$ $[\because C_1\to C_1-C_2]$
$=2\left|\begin{array}{ccc}0& z-x& -x\\ 0& z+x& x\\ y& x& x+y\end{array}\right|$
$=2\left[y\right(xz-x^2+xz+x^2\left)\right]$
$=4xyz=RHS$
Hence proved.