1. **State the problem:** Find the determinant $D$ of the $3\times3$ matrix: $$D = \begin{bmatrix} 1 & 2 & 1 \\ 3 & 1 & 2 \\ 2 & 1 & 4 \end{bmatrix}$$ 2. **Recall the formula for the determinant of a $3\times3$ matrix:** $$\det(D) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the matrix is: $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ 3. **Identify elements:** $$a = 1, b = 2, c = 1$$ $$d = 3, e = 1, f = 2$$ $$g = 2, h = 1, i = 4$$ 4. **Calculate each term:** - $ei - fh = 1 \times 4 - 2 \times 1 = 4 - 2 = 2$ - $di - fg = 3 \times 4 - 2 \times 2 = 12 - 4 = 8$ - $dh - eg = 3 \times 1 - 1 \times 2 = 3 - 2 = 1$ 5. **Compute the determinant:** $$\det(D) = 1 \times 2 - 2 \times 8 + 1 \times 1 = 2 - 16 + 1 = -13$$ 6. **Final answer:** The determinant of matrix $D$ is $\boxed{-13}$. **Note:** The user’s answer was -31, but the correct calculated determinant is -13.