Question:
$\left|\begin{array}{ccc}0 & x y z & x-z \\ y-x & 0 & y-z \\ z-x & z-y & 0\end{array}\right|=$_________
Solution:
$\Delta=\left|\begin{array}{ccc}0 & x y z & x-z \\ y-x & 0 & y-z \\ z-x & z-y & 0\end{array}\right|$
Expanding along $R_{1}$, we get
$=0(0-(y-x)(z-y))-(y-z)(0-(x-z)(z-y))+(z-x)(x y z(y-z)-0)$
$=0-(y-x)(-(x-z)(z-y))+(z-x)(x y z(y-z))$
$=(y-x)(x-z)(z-y)+(z-x) x y z(y-z)$
$=(y-z)(z-x)(y-x+x y z)$
Hence, $\left|\begin{array}{ccc}0 & x y z & x-z \\ y-x & 0 & y-z \\ z-x & z-y & 0\end{array}\right|=\underline{(y-z)(z-x)(y-x+x y z)}$
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