Question

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

Answer 1

\(\Delta = \begin{vmatrix} x &y  &x+y \\   y&x+y  &x \\   x+y&x  &y  \end{vmatrix}\)
Applying R₁\(\rightarrow\)R₁+R₂+R₃, we have
Δ=\(\begin{vmatrix} 2(x+y) &y  &x+y \\   2(x+y)&x+y  &x \\   2(x+y)&x  &y  \end{vmatrix}\)
= 2(x+y)\(\begin{vmatrix} 1&y  &x+y \\   1&x+y  &x \\   1&x  &y  \end{vmatrix}\)
 Applying C₂\(\rightarrow\)C₂-C₁ and C₃\(\rightarrow\)C₃-C₁, we have
Δ=2(x+y)\(\begin{vmatrix} 1 &y  &x+y \\ 0&x  &x \\  0&x-y  & -x \end{vmatrix}\)

Expanding along R₁, we have:
Δ=2(x+y)[-x²+y(x-y)]
=-2(x+y)(x²+y²-yx)
=-2(x³+y³)