1. **Problem:** Solve the determinant of the matrix \(\begin{bmatrix} x - y & y \\ -y & x + y \end{bmatrix}\). 2. **Formula:** The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by \(ad - bc\). 3. **Apply the formula:** \[ \det = (x - y)(x + y) - (y)(-y) = (x - y)(x + y) + y^2 \] 4. **Simplify:** \[ (x - y)(x + y) = x^2 - y^2 \] So, \[ \det = x^2 - y^2 + y^2 = x^2 \] 5. **Conclusion:** The determinant simplifies to \(x^2\). Hence, the determinant of the given matrix is \(x^2\).