Chapter 12.1, Problem 24PPSE

Solution

To determine: The matrix equation to solve the problem by using the given table and explain that to find the total daily production totals at each plant.

The obtained matrix is $\left[\begin{array}{cc}-200& -2200\\ 800& -1000\end{array}\right]$

Given Information:

The table is defined,

	1-color		3-color
	Plastic	Rubber	Plastic	Rubber
Plant 1	500	700	1300	1900
Plant 2	400	1200	600	1600

Explanation:

Consider the given table,

	1-color		3-color
	Plastic	Rubber	Plastic	Rubber
Plant 1	500	700	1300	1900
Plant 2	400	1200	600	1600

First, notice that the table given is per shift, and we have that Plant 11 has 22 shifts, and Plant 22 has 33 shifts.

So if want to make a  $2\times 2$  matrix for both Plants, we need to multiply each element by the number of shifts in each Plant.

The first column is going to represent plastic balls, and the second column is going to represent rubber balls.

Also, the first row is going to represent 11- colour balls and the second row is going to represent 33-colors balls.

Make the metrics of $2\times 2$

  $\begin{array}{c}\text{Plant }1=2\cdot \left[\begin{array}{cc}500& 700\\ 1300& 1900\end{array}\right]\\ =\left[\begin{array}{cc}1000& 1400\\ 2600& 3800\end{array}\right]\end{array}$

Now we need to do the same for Plant 22. Only now, we need to multiply a matrix by 33.

  $\begin{array}{c}\text{Plant 2}=3\cdot \left[\begin{array}{cc}400& 1200\\ 600& 1600\end{array}\right]\\ =\left[\begin{array}{cc}1200& 3600\\ 1800& 4800\end{array}\right]\end{array}$

The last thing we need to do is to subtract these two given matrices to find the result.

  $\begin{array}{c}\text{Plant }1-\text{Plant 2}=\left[\begin{array}{cc}1000& 1400\\ 2600& 3800\end{array}\right]-\left[\begin{array}{cc}1200& 3600\\ 1800& 4800\end{array}\right]\\ =\left[\begin{array}{cc}1000-1200& 1400-3600\\ 2600-1800& 3800-4800\end{array}\right]\\ =\left[\begin{array}{cc}-200& -2200\\ 800& -1000\end{array}\right]\end{array}$

Therefore, the obtained result is $\left[\begin{array}{cc}-200& -2200\\ 800& -1000\end{array}\right]$