Matrix Multiplication 943Be7

1. **Stating the problem:** We are asked to multiply two matrices $$A$$ and $$B$$ where $$A = \begin{bmatrix} 2 & -1 & 2 \\ 3 & -2 & 1 \end{bmatrix}$$ and $$B = \begin{bmatrix} -1 & 2 \\ -3 & 1 \\ 0 & -2 \end{bmatrix}$$. 2. **Formula 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 calculated as: $$$(AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj}$$$ 3. **Check dimensions:** $$A$$ is $$2 \times 3$$ and $$B$$ is $$3 \times 2$$, so the product $$AB$$ is defined and will be a $$2 \times 2$$ matrix. 4. **Calculate each element of $$AB$$:** - Element at position (1,1): $$$(AB)_{11} = 2 \times (-1) + (-1) \times (-3) + 2 \times 0 = -2 + 3 + 0 = 1$$$ - Element at position (1,2): $$$(AB)_{12} = 2 \times 2 + (-1) \times 1 + 2 \times (-2) = 4 - 1 - 4 = -1$$$ - Element at position (2,1): $$$(AB)_{21} = 3 \times (-1) + (-2) \times (-3) + 1 \times 0 = -3 + 6 + 0 = 3$$$ - Element at position (2,2): $$$(AB)_{22} = 3 \times 2 + (-2) \times 1 + 1 \times (-2) = 6 - 2 - 2 = 2$$$ 5. **Final answer:** $$AB = \begin{bmatrix} 1 & -1 \\ 3 & 2 \end{bmatrix}$$ This is the product of matrices $$A$$ and $$B$$.