1. **State the problem:** We are given two matrices: Matrix A (3x3): $$A = \begin{bmatrix} 0.40 & 0.30 & 0.70 \\ 0.30 & 0.60 & 0.25 \\ 0.30 & 0.10 & 0.05 \end{bmatrix}$$ Matrix B (3x2): $$B = \begin{bmatrix} 5000 & 7000 \\ 12000 & 13000 \\ 13000 & 16000 \end{bmatrix}$$ We need to calculate the product $$AB$$. 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. **Calculate each element of $$AB$$:** - Element (1,1): $$0.40 \times 5000 + 0.30 \times 12000 + 0.70 \times 13000 = 2000 + 3600 + 9100 = 14700$$ - Element (1,2): $$0.40 \times 7000 + 0.30 \times 13000 + 0.70 \times 16000 = 2800 + 3900 + 11200 = 17900$$ - Element (2,1): $$0.30 \times 5000 + 0.60 \times 12000 + 0.25 \times 13000 = 1500 + 7200 + 3250 = 11950$$ - Element (2,2): $$0.30 \times 7000 + 0.60 \times 13000 + 0.25 \times 16000 = 2100 + 7800 + 4000 = 13900$$ - Element (3,1): $$0.30 \times 5000 + 0.10 \times 12000 + 0.05 \times 13000 = 1500 + 1200 + 650 = 3350$$ - Element (3,2): $$0.30 \times 7000 + 0.10 \times 13000 + 0.05 \times 16000 = 2100 + 1300 + 800 = 4200$$ 4. **Final product matrix $$AB$$:** $$AB = \begin{bmatrix} 14700 & 17900 \\ 11950 & 13900 \\ 3350 & 4200 \end{bmatrix}$$ This matrix represents the weighted counts for each age group and gender based on political affiliation proportions.