1. The problem is to find the determinant of the matrix $$\begin{pmatrix} x-2 & 0 & 0 \\ 0 & y & -x \\ -y & 0 & z \end{pmatrix}$$ 2. The determinant of a 3x3 matrix $$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ is given by the formula $$\det = a(ei - fh) - b(di - fg) + c(dh - eg)$$ 3. Applying this formula to our matrix, we identify: $$a = x-2, b = 0, c = 0$$ $$d = 0, e = y, f = -x$$ $$g = -y, h = 0, i = z$$ 4. Substitute into the determinant formula: $$\det = (x-2)(y \cdot z - (-x) \cdot 0) - 0(0 \cdot z - (-x)(-y)) + 0(0 \cdot 0 - y(-y))$$ 5. Simplify the terms: $$\det = (x-2)(yz - 0) - 0 + 0 = (x-2)yz$$ 6. Therefore, the determinant of the matrix is: $$\boxed{(x-2)yz}$$