Question

Calculate

\[ \begin{vmatrix} x & y & x + y \\ y & x + y & x \\ x + y & x & y \end{vmatrix} \] 

• A. \( x^3 + y^3 \)
• B. \( x^3 + y^3 + 3x^2y + 3xy^2 + 1 \)
• C. \( -2(x^3 + y^3) \)
• D. \( 2(x^3 + y^3) \)

Hint:

To evaluate a determinant, use cofactor expansion or apply the properties of determinants.

Answer 1

The Correct Option is C

Using the formula for a 3×3 determinant:

\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Applying this to our matrix:

\[ \begin{aligned} &\begin{vmatrix} x & y & x + y \\ y & x + y & x \\ x + y & x & y \end{vmatrix} \\ &= x\left[(x + y)(y) - (x)(x)\right] \\ &\quad - y\left[y(y) - (x)(x + y)\right] \\ &\quad + (x + y)\left[y(x) - (x + y)(x + y)\right] \\ &= x(xy + y^2 - x^2) \\ &\quad - y(y^2 - x^2 - xy) \\ &\quad + (x + y)(xy - x^2 - 2xy - y^2) \\ &= x^2y + xy^2 - x^3 - y^3 + x^2y + xy^2 \\ &\quad - x^3 - 2x^2y - 2xy^2 - y^3 \\ &= -2x^3 - 2y^3 \\ &= -2(x^3 + y^3) \end{aligned} \]

The correct answer is \(\boxed{c}\) \(-2(x^3 + y^3)\).