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4.07 Further applications of matrices

Lesson

We have seen examples of matrices used to solve problems across many real-world applications. Such as:

  • Systems relating to economics (prices, profits, losses, markups)
  • Communication and traffic networks (connectivity, number of routes)
  • Storage and analysis of data

Some key things to remember:

  • If we have to convert a system into a matrix, check what is placed in rows and columns. The operations we can meaningfully carry out on matrices rely on us understanding the context.
  • If we need to multiply, set up your matrices such that the dimensions allow multiplication and that the multiplication makes sense in the context of the question.
  • Always check that the answer seems reasonable.

Matrices allow us to perform multiple individual calculations in one operation and can be used on very large arrays of data. As such, matrix applications are wide reaching, let's look at some further applications.

Practice questions

question 1

A second-hand bookstore sells textbooks at a markup of $50%$50%. The table shows the amounts they paid for old textbooks during the past academic year.

  Semester 1 Semester 2
Business $\$940$$940 $\$980$$980
Law $\$1020$$1020 $\$1170$$1170
Mathematics $\$930$$930 $\$1160$$1160
Science $\$1180$$1180 $\$1040$$1040
Engineering $\$1150$$1150 $\$970$$970
  1. Organise the purchase costs into a cost matrix, with each row representing a subject and columns representing semesters.

    $C$C $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
  2. Organise the revenue that will be generated when they manage to sell all the textbooks into a revenue matrix.

    $R$R $=$= $\editable{}$
        $940$940 $980$980    
        $1020$1020 $1170$1170    
        $930$930 $1160$1160    
        $1180$1180 $1040$1040    
        $1150$1150 $970$970    
      $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
  3. Complete the profit matrix.

    $P$P $=$=
        $1410$1410 $1470$1470    
        $1530$1530 $1755$1755    
        $1395$1395 $1740$1740    
        $1770$1770 $1560$1560    
        $1725$1725 $1455$1455    
    $-$
        $940$940 $980$980    
        $1020$1020 $1170$1170    
        $930$930 $1160$1160    
        $1180$1180 $1040$1040    
        $1150$1150 $970$970    
      $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
  4. How much profit would the bookstore have generated from the sale of all these textbooks?

question 2

The map shows the roads that connect three towns.

  1. Complete $A$A, a matrix showing the number of direct $1$1-step paths from each town to another town.

    $A$A $=$=
      To:   Basstown Cowra Dinerwa    
      Basstown   $\editable{}$ $\editable{}$ $\editable{}$    
    From: Cowra   $\editable{}$ $\editable{}$ $\editable{}$    
      Dinerwa   $\editable{}$ $\editable{}$ $\editable{}$    
                   
  2. Find $A^2$A2.

    $A^2$A2 $=$=   $0$0 $0$0 $1$1         $0$0 $0$0 $1$1    
    $0$0 $0$0 $1$1 $0$0 $0$0 $1$1
    $1$1 $1$1 $0$0 $1$1 $1$1 $0$0
     
      $=$=   $\editable{}$ $\editable{}$ $\editable{}$    
    $\editable{}$ $\editable{}$ $\editable{}$
    $\editable{}$ $\editable{}$ $\editable{}$
  3. How many ways are there to go from Cowra to Dinerwa with one stop in between?

question 3

The matrix $H$H below shows three friends and their scores over four games played against each other.

If $A$A and $B$B are matrices of ones as shown, answer the following questions.

$H$H $=$=
      Game 1 Game 2 Game 3 Game 4    
  Mandy   $10$10 $12$12 $15$15 $20$20    
  Millie   $8$8 $20$20 $6$6 $14$14    
  Max   $11$11 $16$16 $15$15 $18$18    
                 
$A$A $=$=
             
    $1$1 $1$1 $1$1    
             
  $B$B $=$=
    $1$1    
    $1$1    
    $1$1    
    $1$1    
  1. Which calculation will give the average number of points that each player scored?

    $\frac{1}{3}AH$13AH

    A

    $\frac{1}{4}AH$14AH

    B

    $\frac{1}{3}HB$13HB

    C

    $\frac{1}{4}HB$14HB

    D
  2. Who had the highest average score overall?

    Mandy

    A

    Millie

    B

    Max

    C

question 4

A company sells five different models of coffee machines through three different outlets. The stock on hand and corresponding wholesale and retail prices for each model and outlet are shown below.

Model of coffee machine $A$A $B$B $C$C $D$D $E$E
JB Hi Fi $2$2 $4$4 $2$2 $5$5 $1$1
Harvey Norman $1$1 $2$2 $3$3 $4$4 $2$2
Good Guys $6$6 $0$0 $4$4 $3$3 $2$2
Wholesale price ($\$$$) $90$90 $100$100 $110$110 $150$150 $190$190
Retail Price ($\$$$) $99$99 $120$120 $150$150 $220$220 $300$300
  1. Complete matrix $S$S below, which shows the stock available in each store.

    $S$S $=$=
          $A$A $B$B $C$C $D$D $E$E    
      JB Hi Fi   $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$    
      Harvey Norman   $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$    
      Good guys   $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$    
                       
  2. If matrix $T$T is to represent the wholesale and retail prices of each coffee machine, and we want to be able to multiply it by matrix $S$S later, what dimensions should it have?

     

    Matrix $T$T will be a $\editable{}$$\times$×$\editable{}$ matrix.

  3. Complete matrix $T$T below.

    $T$T $=$=
          Wholesale Retail    
      $A$A   $\editable{}$ $\editable{}$    
      $B$B   $\editable{}$ $\editable{}$    
      $C$C   $\editable{}$ $\editable{}$    
      $D$D   $\editable{}$ $\editable{}$    
      $E$E   $\editable{}$ $\editable{}$    
                 
  4. Complete the matrix multiplication of $S$S by $T$T to produce matrix $P$P:

    $P$P $=$=
        $2$2 $4$4 $2$2 $5$5 $1$1    
        $1$1 $2$2 $3$3 $4$4 $2$2    
        $6$6 $0$0 $4$4 $3$3 $2$2    
    $\times$×
        $90$90 $99$99    
        $100$100 $120$120    
        $110$110 $150$150    
        $150$150 $220$220    
        $190$190 $300$300    
      $=$=
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
        $\editable{}$ $\editable{}$    
       
  5. Which of the following is a correct interpretation of matrix $P$P?

    The total amount of money each store will make when every coffee machine is sold.

    A

    The total amount of coffee machine that can be sold at the wholesale and retail prices.

    B

    The total wholesale and retail dollars each store stands to receive when every coffee machine is sold.

    C

Outcomes

ACMGM013

use matrices for storing and displaying information that can be presented in rows and columns; for example, databases, links in social or road networks

ACMGM016

use matrices, including matrix products and powers of matrices, to model and solve problems; for example, costing or pricing problems, squaring a matrix to determine the number of ways pairs of people in a communication network can communicate with each other via a third person

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