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This is probably a simple question

A factory produce a good (1) that requires 3 labor-hours in the assembly department and 1 labor-hour in the finishing department. Assembly personnel receive 19 per hour and finishing personnel receive 16 per hour.

I need to write a matrix or vector product that would give the total labor cost per good 1.

$\displaystyle =\begin{bmatrix} 3 & 1\end{bmatrix}$$\displaystyle $$\displaystyle \begin{bmatrix} 19\\ 16\end{bmatrix}$

(Thanks SiongthyeGoh)

Furthermore, the company now produces two goods (1 & 2). Good (2) requires 5 labor-hours in the assembly department and 1.5 labor-hours in the finishing department.

Labour-hour information can be combined into a single matrix

$L =\displaystyle \begin{bmatrix} 5&1.5\\ 3&1\end{bmatrix}$

The company has two manufacturing plants San Diego, and Los Angeles. The hourly rates for each department are given by

$H =\displaystyle \begin{bmatrix} 22&19\\ 18&16\end{bmatrix}$

where the assembly and finishing departments are the rows and San Diego and Los Angeles are the columns. Identify a matrix product that would yield the total labor cost for each type of good and evaluate.

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I'd write your labor information matrix as $$L=\left[\matrix{3&1\cr 5&1.5\cr}\right]\ ,$$ so that the first row refers to good$_1$ and the second row to good$_2$. Similarly, you have a plant$_1$ and a plant$_2$ whose labor costs are entered into the first and second columns of $$H=\left[\matrix{22&19\cr18&16\cr}\right]\ .$$ If you now compute the product $$P:=L\>H=\left[\matrix{84&73\cr 137&119\cr}\right]$$ then the entry $$P_{ik}=L_{i1}H_{1k}+L_{i2}H_{2k}\qquad(1\leq i\leq2, \ 1\leq k\leq2)$$ tells you how much it costs to produce $1$ unit of good$_i$ at plant$_k\>$.

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You want to multiply 3 with 19 and 1 with 16, and then sum it up. A convenient way to express it is

$$\begin{bmatrix} 3 & 1\end{bmatrix}\begin{bmatrix} 19 \\ 16 \end{bmatrix}$$

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