Question

Using cofactors of elements of third column , evaluate $△=\left|\begin{array}{ccc}1& x& yz\\ 1& y& zx\\ 1& z& xy\end{array}\right|$

Answer 1

$△=\left|\begin{array}{ccc}1& x& yz\\ 1& y& zx\\ 1& z& xy\end{array}\right|$

Cofactors of $3rd$ column

$△=\left|\begin{array}{ccc}1& x& z-y\\ 1& y& z-x\\ 1& z& y-x\end{array}\right|△=\left|\begin{array}{ccc}1& x& z\\ 1& y& z\\ 1& z& y\end{array}\right|-\left|\begin{array}{ccc}1& x& y\\ 1& y& x\\ 1& z& x\end{array}\right|$

$=1(y^2-z^2)-x(y-z)+z(z-y)-1(xy-xz)+x(x-x)-y(z-y)$

$=0$