1. **Problem Statement:** Find the product of two 3x3 matrices: $$A = \begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}, \quad B = \begin{bmatrix}9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1\end{bmatrix}$$ 2. **Formula:** The product matrix $C = A \times B$ is given by: $$c_{ij} = \sum_{k=1}^3 a_{ik} b_{kj}$$ where $c_{ij}$ is the element in the $i$th row and $j$th column of $C$. 3. **Step-by-step multiplication:** - Calculate $c_{11}$: $$c_{11} = 1 \times 9 + 2 \times 6 + 3 \times 3 = 9 + 12 + 9 = 30$$ - Calculate $c_{12}$: $$c_{12} = 1 \times 8 + 2 \times 5 + 3 \times 2 = 8 + 10 + 6 = 24$$ - Calculate $c_{13}$: $$c_{13} = 1 \times 7 + 2 \times 4 + 3 \times 1 = 7 + 8 + 3 = 18$$ - Calculate $c_{21}$: $$c_{21} = 4 \times 9 + 5 \times 6 + 6 \times 3 = 36 + 30 + 18 = 84$$ - Calculate $c_{22}$: $$c_{22} = 4 \times 8 + 5 \times 5 + 6 \times 2 = 32 + 25 + 12 = 69$$ - Calculate $c_{23}$: $$c_{23} = 4 \times 7 + 5 \times 4 + 6 \times 1 = 28 + 20 + 6 = 54$$ - Calculate $c_{31}$: $$c_{31} = 7 \times 9 + 8 \times 6 + 9 \times 3 = 63 + 48 + 27 = 138$$ - Calculate $c_{32}$: $$c_{32} = 7 \times 8 + 8 \times 5 + 9 \times 2 = 56 + 40 + 18 = 114$$ - Calculate $c_{33}$: $$c_{33} = 7 \times 7 + 8 \times 4 + 9 \times 1 = 49 + 32 + 9 = 90$$ 4. **Final product matrix:** $$C = \begin{bmatrix}30 & 24 & 18 \\ 84 & 69 & 54 \\ 138 & 114 & 90\end{bmatrix}$$ This matrix multiplication combines rows of $A$ with columns of $B$ to produce $C$. 5. **Summary:** Matrix multiplication is not element-wise but involves summing products of corresponding row and column elements.