Chapter 7.2, Problem 50E

Solution

a.

To determine: The matrix $B$ of order $1\times 3$ that represent the number of each type of house to be built.

  $\begin{array}{l}\text{ Steel Wood Galss Paint Labor}\\ R=\begin{array}{c}\text{Ranch}\\ \text{Cape Cod}\\ \text{Colonial}\end{array}\left[\begin{array}{ccccc}5& 22& 14& 7& 17\\ 7& 20& 10& 9& 21\\ 6& 27& 8& 5& 13\end{array}\right]\end{array}$

The given identity is true.

Given Information:

The matrix is defined as,

  $\begin{array}{l}\text{ Steel Wood Galss Paint Labor}\\ R=\begin{array}{c}\text{Ranch}\\ \text{Cape Cod}\\ \text{Colonial}\end{array}\left[\begin{array}{ccccc}5& 22& 14& 7& 17\\ 7& 20& 10& 9& 21\\ 6& 27& 8& 5& 13\end{array}\right]\end{array}$

And Construction A building contractor has agreed to build six ranch-style houses, seven Cape Cod-style houses, and 14 colonial-style houses. The numbers of units of raw materials that go into each type of house are shown in matrix R .

Explanation:

Consider the given matrix,

  $\begin{array}{l}\text{ Steel Wood Galss Paint Labor}\\ R=\begin{array}{c}\text{Ranch}\\ \text{Cape Cod}\\ \text{Colonial}\end{array}\left[\begin{array}{ccccc}5& 22& 14& 7& 17\\ 7& 20& 10& 9& 21\\ 6& 27& 8& 5& 13\end{array}\right]\end{array}$

It is know that there are three type of material making house and A building contractor has agreed to build six ranch-style houses, seven Cape Cod-style houses, and 14 colonial-style houses.

Write them in matrix form.

  $B=\left[\begin{array}{ccc}\text{Ranch}& \text{Cape Cod}& \text{Colonial}\\ 6& 7& 14\end{array}\right]$

Therefore, the required matrix is $B=\left[\begin{array}{ccc}\text{Ranch}& \text{Cape Cod}& \text{Colonial}\\ 6& 7& 14\end{array}\right]$ .

b.

To determine: The matrix product that gives the number of units of each row material needed to build the houses.

The obtained matrix is $A=\left[\begin{array}{ccccc}163& 650& 266& 175& 431\end{array}\right]$ .

Given Information:

The matrix is defined as,

  $\begin{array}{l}\text{ Steel Wood Galss Paint Labor}\\ R=\begin{array}{c}\text{Ranch}\\ \text{Cape Cod}\\ \text{Colonial}\end{array}\left[\begin{array}{ccccc}5& 22& 14& 7& 17\\ 7& 20& 10& 9& 21\\ 6& 27& 8& 5& 13\end{array}\right]\end{array}$

And,

  $B=\left[\begin{array}{ccc}6& 7& 14\end{array}\right]$

Explanation:

Consider the given matrix,

  $\begin{array}{l}\text{ Steel Wood Galss Paint Labor}\\ R=\begin{array}{c}\text{Ranch}\\ \text{Cape Cod}\\ \text{Colonial}\end{array}\left[\begin{array}{ccccc}5& 22& 14& 7& 17\\ 7& 20& 10& 9& 21\\ 6& 27& 8& 5& 13\end{array}\right]\end{array}$

And,

  $B=\left[\begin{array}{ccc}6& 7& 14\end{array}\right]$

Now, multiply both the matrix.

  $\begin{array}{c}D=\left[\begin{array}{ccc}6& 7& 14\end{array}\right]\cdot \left[\begin{array}{ccccc}5& 22& 14& 7& 17\\ 7& 20& 10& 9& 21\\ 6& 27& 8& 5& 13\end{array}\right]\\ =\left[\begin{array}{ccccc}\left(\begin{array}{l}30+49\\ +84\end{array}\right)& \left(\begin{array}{l}132+140\\ +378\end{array}\right)& \left(\begin{array}{l}84+70\\ +112\end{array}\right)& \left(\begin{array}{l}42+63\\ +70\end{array}\right)& \left(\begin{array}{l}102+147\\ +182\end{array}\right)\end{array}\right]\\ =\left[\begin{array}{ccccc}163& 650& 266& 175& 431\end{array}\right]\end{array}$

Therefore, the obtained matrix is $B=\left[\begin{array}{ccccc}163& 650& 266& 175& 431\end{array}\right]$ .

c.

To determine: The matrix C of $5\times 1$ that represents per unit cost of each type of row materiel.

The obtained matrix is $C=\left[\begin{array}{cc}\text{Steel}& 1600\\ \text{wood}& 900\\ \text{glass}& 500\\ \text{Paint}& 100\\ \text{Labor}& 1000\end{array}\right]$ .

Given Information:

Refer the matrix form the previous parts.

Explanation:

Consider the given information,

Assume that steel costs $1600 a unit, wood $900 a unit, glass $500 a unit, paint $100 a unit, and labour $1000 a unit.

Now, make the cost matrix.

  $C=\left[\begin{array}{cc}\text{Steel}& 1600\\ \text{wood}& 900\\ \text{glass}& 500\\ \text{Paint}& 100\\ \text{Labor}& 1000\end{array}\right]$

Therefore, the above matrix is the required matrix.

d.

To determine: The matrix product that givens the cost of each house.

The result is $RC=\left[\begin{array}{c}52500\\ 56100\\ 51400\end{array}\right]$ .

Given Information:

Refer the previous part matrix.

Explanation:

Consider the given identity,

The matrix R and C is defined as,

  $\begin{array}{l}\text{ Steel Wood Galss Paint Labor}\\ R=\begin{array}{c}\text{Ranch}\\ \text{Cape Cod}\\ \text{Colonial}\end{array}\left[\begin{array}{ccccc}5& 22& 14& 7& 17\\ 7& 20& 10& 9& 21\\ 6& 27& 8& 5& 13\end{array}\right]\end{array}$

Now, calculate the matrix multiplication $RC$ .

  $\begin{array}{c}RC=\left[\begin{array}{ccccc}5& 22& 14& 7& 17\\ 7& 20& 10& 9& 21\\ 6& 27& 8& 5& 13\end{array}\right]\left[\begin{array}{c}1600\\ 900\\ 500\\ 100\\ 1000\end{array}\right]\\ =\left[\begin{array}{c}8000+14400+7000+700+17000\\ 11200+18000+5000+900+21000\\ 9600+24300+4000+500+13000\end{array}\right]\\ =\left[\begin{array}{c}52500\\ 56100\\ 51400\end{array}\right]\end{array}$

Therefore, the obtained matrix is $RC=\left[\begin{array}{c}52500\\ 56100\\ 51400\end{array}\right]$ .

e.

To determine: The product of $BRC$ and explain that information represented by it.

The result is $BRC=\left[1427300\right]$ .

Given Information:

Refer the previous part matrix.

Explanation:

Consider the given matrix,

  $RC=\left[\begin{array}{c}52500\\ 56100\\ 51400\end{array}\right]$ and $B=\left[\begin{array}{ccc}6& 7& 14\end{array}\right]$

Now, calculate the matrix multiplication $BRC$ .

  $\begin{array}{c}BRC=\left[\begin{array}{ccc}6& 7& 14\end{array}\right]\left[\begin{array}{c}52500\\ 56100\\ 51400\end{array}\right]\\ =\left[163\left(1600\right)+650\left(900\right)+175\left(100\right)+431\left(1000\right)\right]\\ =\left[1427300\right]\end{array}$

And this is representing that this is the contractor's total cost to build all the houses.

Therefore, the obtained matrix is $BRC=\left[1427300\right]$ .