$\left|\begin{matrix}1&1&1\\1&1+x&1\\1&1&1+y\\\end{matrix}\right|=?$

$\Delta=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\mathrm{x} & 1 \\ 1 & 1 & 1+\mathrm{y} \end{array}\right|$

$\mathrm{R}_{1}=\mathrm{R}_{1}-\mathrm{R}_{2}$

$\Rightarrow \Delta=\left|\begin{array}{ccc} 0 & -\mathrm{x} & 0 \\ 1 & 1+\mathrm{x} & 1 \\ 1 & 1 & 1+\mathrm{y} \end{array}\right|$

Expanding along $\mathrm{R}_{1}$

$\begin{array}{l} \Rightarrow \Delta=0-(-\mathrm{x})\{1+\mathrm{y}-1\}+0 \\ \Rightarrow \Delta=\mathrm{xy} \end{array}$