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)