1. **Problem Statement:** Multiply the two matrices $$A = \begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & 2 \\ 1 & 3 & 1 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 2 & 2 & 0 \\ 1 & 3 & 2 \\ 3 & 2 & 0 \end{pmatrix}$$ 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 $$ (AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj} $$ 3. **Calculate each element of the product matrix $AB$: ** - First row, first column: $$1 \times 2 + 0 \times 1 + 3 \times 3 = 2 + 0 + 9 = 11$$ - First row, second column: $$1 \times 2 + 0 \times 3 + 3 \times 2 = 2 + 0 + 6 = 8$$ - First row, third column: $$1 \times 0 + 0 \times 2 + 3 \times 0 = 0 + 0 + 0 = 0$$ - Second row, first column: $$2 \times 2 + 1 \times 1 + 2 \times 3 = 4 + 1 + 6 = 11$$ - Second row, second column: $$2 \times 2 + 1 \times 3 + 2 \times 2 = 4 + 3 + 4 = 11$$ - Second row, third column: $$2 \times 0 + 1 \times 2 + 2 \times 0 = 0 + 2 + 0 = 2$$ - Third row, first column: $$1 \times 2 + 3 \times 1 + 1 \times 3 = 2 + 3 + 3 = 8$$ - Third row, second column: $$1 \times 2 + 3 \times 3 + 1 \times 2 = 2 + 9 + 2 = 13$$ - Third row, third column: $$1 \times 0 + 3 \times 2 + 1 \times 0 = 0 + 6 + 0 = 6$$ 4. **Final product matrix:** $$AB = \begin{pmatrix} 11 & 8 & 0 \\ 11 & 11 & 2 \\ 8 & 13 & 6 \end{pmatrix}$$ This completes the matrix multiplication step-by-step.