1. **State the problem:** We have two factories $F_1$ and $F_2$ selling sofas (S), chairs (C), and tables (T) with profits per item of 450000, 100000, and 350000 respectively. Sales quantities are given in a matrix. We want to find the total profit using matrices and then evaluate it. 2. **Set up matrices:** Let $\mathbf{S}$ be the sales matrix and $\mathbf{P}$ be the profit vector: $$\mathbf{S} = \begin{bmatrix} 100 & 400 & 700 \\ 900 & 300 & 500 \end{bmatrix}, \quad \mathbf{P} = \begin{bmatrix} 450000 \\ 100000 \\ 350000 \end{bmatrix}$$ 3. **Matrix multiplication for total profit:** Total profit for each factory is given by the product $\mathbf{S} \cdot \mathbf{P}$: $$\mathbf{T} = \mathbf{S} \cdot \mathbf{P} = \begin{bmatrix} 100 & 400 & 700 \\ 900 & 300 & 500 \end{bmatrix} \cdot \begin{bmatrix} 450000 \\ 100000 \\ 350000 \end{bmatrix}$$ 4. **Calculate each element:** For $F_1$: $$100 \times 450000 + 400 \times 100000 + 700 \times 350000 = 45000000 + 40000000 + 245000000 = 330000000$$ For $F_2$: $$900 \times 450000 + 300 \times 100000 + 500 \times 350000 = 405000000 + 30000000 + 175000000 = 610000000$$ 5. **Interpretation:** The total profit from $F_1$ is 330000000 and from $F_2$ is 610000000. **Final answer:** $$\mathbf{T} = \begin{bmatrix} 330000000 \\ 610000000 \end{bmatrix}$$