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

Let $\Delta = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&x&{yz}\\1&y&{zx}\\1&z&{xy}\end{array}} \right|,$

Cofactors of elements of third column are
${A_{13}} = {( - 1)^{1 + 3}}\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&y\\1&z\end{array}} \right| = z - y,{A_{23}} = {( - 1)^{2 + 3}}\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&x\\1&z\end{array}} \right| = - (z - x)$

${A_{33}} = {( - 1)^{3 + 3}}\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&x\\1&y\end{array}} \right| = y - x = - (x - y)$

Now,
$\Delta = {a_{13}}{A_{13}} + {a_{23}}{A_{23}} + {a_{33}}{A_{33}}$

$\Delta = - yz(y - z) - zx(z - x) - xy(x - y)$

$= zy(z - y) + zx(x - z) + xy(y - x)$

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

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

$= {x^2}(z - y) + x({y^2} - {z^2}) + yz(z - y)$

$= (z - y)[{x^2} - x(y + z) + yz] = (z - y)\{ x(x - y) - z(x - y)\}$
$= (z - y)(x - y)(x - z) = (x - y)(y - z)(z - x)$