1. **Problem Statement:** We have two factory outlets F₁ and F₂ selling sofas (S), chairs (C), and tables (T) with profits per item TZS 450000, 100000, and 350000 respectively. 2. **Given Data:** Sales matrix \( A = \begin{bmatrix} 100 & 400 & 700 \\ 900 & 300 & 500 \end{bmatrix} \) where rows represent outlets F₁ and F₂, and columns represent quantities of sofas, chairs, and tables sold. Profit per item matrix \( P = \begin{bmatrix} 450000 \\ 100000 \\ 350000 \end{bmatrix} \). 3. **Expression for Total Profit:** Total profit for each outlet is the matrix product \( A \times P \). 4. **Calculation:** \[ \begin{aligned} A \times P &= \begin{bmatrix} 100 & 400 & 700 \\ 900 & 300 & 500 \end{bmatrix} \times \begin{bmatrix} 450000 \\ 100000 \\ 350000 \end{bmatrix} \\ &= \begin{bmatrix} 100 \times 450000 + 400 \times 100000 + 700 \times 350000 \\ 900 \times 450000 + 300 \times 100000 + 500 \times 350000 \end{bmatrix} \\ &= \begin{bmatrix} 45000000 + 40000000 + 245000000 \\ 405000000 + 30000000 + 175000000 \end{bmatrix} \\ &= \begin{bmatrix} 330000000 \\ 610000000 \end{bmatrix} \end{aligned} \] 5. **Interpretation:** The total profit from outlet F₁ is TZS 330000000 and from outlet F₂ is TZS 610000000. **Final Answer:** Total profit vector \( \begin{bmatrix} 330000000 \\ 610000000 \end{bmatrix} \).