1. **State the problem:** Evaluate the determinant of the matrix $$D = \begin{vmatrix} 1 & 2 & -1 \\ 3 & 0 & 0 \\ 2 & -2 & -1 \end{vmatrix}$$ 2. **Formula for determinant of a 3x3 matrix:** $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the matrix is $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ 3. **Identify elements:** $$a=1, b=2, c=-1, d=3, e=0, f=0, g=2, h=-2, i=-1$$ 4. **Calculate each part:** $$ei - fh = (0)(-1) - (0)(-2) = 0 - 0 = 0$$ $$di - fg = (3)(-1) - (0)(2) = -3 - 0 = -3$$ $$dh - eg = (3)(-2) - (0)(2) = -6 - 0 = -6$$ 5. **Substitute into determinant formula:** $$D = 1 \times 0 - 2 \times (-3) + (-1) \times (-6)$$ 6. **Simplify:** $$D = 0 + 6 + 6 = 12$$ 7. **Check for cancellation or errors:** The calculation is correct; the determinant is 12. **Final answer:** The determinant $D = 12$. Note: The options given (4, 16, 24, -16) do not include 12, so the determinant is 12 based on calculation.