1. **Problem Statement:** Calculate the product of the two matrices: $$ \left(\begin{array}{ccc} 1 & 0 & 3 \\ 2 & 1 & 2 \\ 1 & 3 & 1 \end{array}\right) \times \left(\begin{array}{ccc} 2 & 2 & 0 \\ 1 & 3 & 2 \\ 3 & 2 & 0 \end{array}\right). $$ 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 computed as: $$ (AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj}. $$ 3. **Step-by-step Calculation:** - Calculate each element of the resulting matrix by multiplying rows of the first matrix by columns of the second matrix. Row 1: - Element (1,1): $1\times 2 + 0\times 1 + 3\times 3 = 2 + 0 + 9 = 11$ - Element (1,2): $1\times 2 + 0\times 3 + 3\times 2 = 2 + 0 + 6 = 8$ - Element (1,3): $1\times 0 + 0\times 2 + 3\times 0 = 0 + 0 + 0 = 0$ Row 2: - Element (2,1): $2\times 2 + 1\times 1 + 2\times 3 = 4 + 1 + 6 = 11$ - Element (2,2): $2\times 2 + 1\times 3 + 2\times 2 = 4 + 3 + 4 = 11$ - Element (2,3): $2\times 0 + 1\times 2 + 2\times 0 = 0 + 2 + 0 = 2$ Row 3: - Element (3,1): $1\times 2 + 3\times 1 + 1\times 3 = 2 + 3 + 3 = 8$ - Element (3,2): $1\times 2 + 3\times 3 + 1\times 2 = 2 + 9 + 2 = 13$ - Element (3,3): $1\times 0 + 3\times 2 + 1\times 0 = 0 + 6 + 0 = 6$ 4. **Final Answer:** $$ \left(\begin{array}{ccc} 11 & 8 & 0 \\ 11 & 11 & 2 \\ 8 & 13 & 6 \end{array}\right). $$ This is the product of the given matrices.