Question

Calculate the values of the determinants:
$\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|R_1=R_1-R_2\Rightarrow \Delta =\left|\begin{array}{ccc}0& -x& 0\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =0-(-x)\{1+y-1\}+0\Rightarrow \Delta =xy$