Question

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

Answer 1

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

$=yz(z-y)-yx(z-x)+xy(y-x)$

$=y{z}^2-y^{2}z-xyz+y{x}^2+x{y}^2-x^{2}y$

$=x{y}^2+y{z}^2{y}^{2}z-xyz$