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 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