1. **State the problem:** Multiply the matrices $$A = \begin{bmatrix}-2 & -4 & 0 \\ 0 & 4 & -1\end{bmatrix}$$ and $$B = \begin{bmatrix}0 & 2 & -2 \\ -1 & -1 & 1 \\ 1 & -3 & -1\end{bmatrix}$$ 2. **Recall the rule for matrix multiplication:** If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then the product $AB$ is an $m \times p$ matrix where each element is computed as $$ (AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj} $$ 3. **Calculate each element of the product matrix $C = AB$: ** - $C_{11} = (-2)(0) + (-4)(-1) + (0)(1) = 0 + 4 + 0 = 4$ - $C_{12} = (-2)(2) + (-4)(-1) + (0)(-3) = -4 + 4 + 0 = 0$ - $C_{13} = (-2)(-2) + (-4)(1) + (0)(-1) = 4 - 4 + 0 = 0$ - $C_{21} = (0)(0) + (4)(-1) + (-1)(1) = 0 - 4 - 1 = -5$ - $C_{22} = (0)(2) + (4)(-1) + (-1)(-3) = 0 - 4 + 3 = -1$ - $C_{23} = (0)(-2) + (4)(1) + (-1)(-1) = 0 + 4 + 1 = 5$ 4. **Write the resulting matrix:** $$C = \begin{bmatrix}4 & 0 & 0 \\ -5 & -1 & 5\end{bmatrix}$$ **Final answer:** $$\boxed{\begin{bmatrix}4 & 0 & 0 \\ -5 & -1 & 5\end{bmatrix}}$$