Evaluate $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&x&y\\1&{x + y}&y\\1&x&{x + y}\end{array}} \right|$

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

Applying ${R_2} \to {R_2} - {R_1}$ and ${R_3} \to {R_3} - {R_1},$

we get
$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&x&y\\0&y&0\\0&0&x\end{array}} \right|$

Expanding along ${C_1}$,

we get
$1 \times y \times x = xy.$
Using properties of determinants in questions 11 to15, prove that: