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 |
Organise the purchase costs into a cost matrix, with each row representing a subject and columns representing semesters.
$C$C $=$= $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 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 $=$= $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 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 $=$= $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 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.
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 $\editable{}$ $\editable{}$ $\editable{}$ 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 $=$= $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ How many ways are there to go from Cowra to Dinerwa with one stop in between?
question 3
Jack and John are planning a fishing trip. Jack needs $2$2 sinkers, $3$3 packs of hooks and $4$4 jigs, while John needs $1$1 sinker, $4$4 packs of hooks and $2$2 jigs.
The prices at Fish Tackle are $\$1.20$$1.20 per sinker, $\$4.00$$4.00 for a pack of hooks, and jigs are $\$2.20$$2.20 each. At Tackle Bite, sinkers are $\$1.90$$1.90 each, packs of hooks are $\$4.50$$4.50, and jigs are $\$0.90$$0.90 each.
Display the required items for Jack and John in a $3\times2$3×2 matrix, $W$W.
Display the prices for these items at the two stores in a $2\times3$2×3 matrix, $P$P.
Which of the following calculations will help Jack and John determine where they should shop?
$P\times W$P×W
A$W\times P$W×P
BCalculate the chosen matrix multiplication.
At which store should they each shop?
question 4
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 | $=$= |
|
| $A$A | $=$= |
|
$B$B | $=$= |
|
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
DWho had the highest average score overall?
Mandy
AMillie
BMax
C
question 5
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 |
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 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 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.
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 $\editable{}$ $\editable{}$ 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 $=$= $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 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.
AThe total amount of coffee machine that can be sold at the wholesale and retail prices.
BThe total wholesale and retail dollars each store stands to receive when every coffee machine is sold.
C