Evaluate : | * 20 c 1&x&y 1& x + y &y 1&x& x + y | - Vidyadip

Question

Evaluate : $\left| {\begin{array}{*{20}{c}} 1&x&y \\ 1&{x + y}&y \\ 1&x&{x + y} \end{array}} \right|$

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Answer

Let $\Delta = \left| {\begin{array}{*{20}{c}} 1&x&y \\ 1&{x + y}&y \\ 1&x&{x + y} \end{array}} \right|$

$\left[ {{R_2} \to {R_2} - {R_1}\,\,and\,\,{R_3} \to {R_3} - {R_1}} \right]$

$= \left| {\begin{array}{*{20}{c}} 1&x&y \\ 0&{x + y - y}&0 \\ 0&0&{x + y - y} \end{array}} \right|$

$ = \left| {\begin{array}{*{20}{c}} 1&x&y \\ 0&y&0 \\ 0&0&x \end{array}} \right|$

Expanding along Ist column

$= 1\left| {\begin{array}{*{20}{c}} y&0 \\ 0&x \end{array}} \right|$

= xy

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