Answer
See below.
Work Step by Step
We know that for a matrix
\[
\left[\begin{array}{rrr}
a & b & c \\
d &e & f \\
g &h & i \\
\end{array} \right]
\]
the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$
Hence here $D=x^2y-x^2z-xy^2+xz^2+y^2z-yz^2=(x-z)(y-z)(x-y)$
Thus we proved what we had to.