1. **State the problem:** Calculate the determinant of the matrix $$A = \begin{bmatrix} 2 & -1 & 0 \\ 2 & 0 & 1 \\ 4 & 3 & 0 \end{bmatrix}$$ (Note: The matrix given in the user message is ambiguous, but assuming a 3x3 matrix with rows as $[2, -1, 0], [2, 0, 1], [4, 3, 0]$ to match the 3x3 format.) 2. **Recall the formula for determinant of a 3x3 matrix:** $$\det A = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the matrix is $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ 3. **Assign values:** $$a=2, b=-1, c=0, d=2, e=0, f=1, g=4, h=3, i=0$$ 4. **Calculate each term:** - $$ei - fh = 0 \times 0 - 1 \times 3 = -3$$ - $$di - fg = 2 \times 0 - 1 \times 4 = -4$$ - $$dh - eg = 2 \times 3 - 0 \times 4 = 6$$ 5. **Compute determinant:** $$\det A = 2 \times (-3) - (-1) \times (-4) + 0 \times 6 = -6 - 4 + 0 = -10$$ 6. **Interpretation:** The determinant of matrix $$A$$ is $$-10$$. **Final answer:** $$\boxed{-10}$$