Question

Evaluate\(\begin{vmatrix} 1 & x & y\\ 1 & x+y & y\\1&x&x+y \end{vmatrix}\)

Answer 1

\(Δ=\)\(\begin{vmatrix} 1 & x & y\\ 1 & x+y & y\\1&x&x+y \end{vmatrix}\)
Applying \(R_2\rightarrow R_2-R_1\) and \(R_3\rightarrow R_3-R_1\),we have
\(Δ=\)\(\begin{vmatrix} 1 & x & y\\ 0 & y & 0\\0&0&x \end{vmatrix}\)

Expanding along \(C_1\),we have:
\(Δ=1(xy-0)=xy\)