Four friends are comparing their bicycles. They have realised that the wheels on the bikes are different sizes. The matrix \begin{bmatrix} 30 & 32 \\ 29 & 28 \end{bmatrix} shows the radii of the wheels on each of their bikes in centimetres.
What scalar multiplier will give the distances travelled by each bicycle in a single rotation of their wheels?
Hence find these distances, to the nearest whole number. Express your answer as a \\ 2 \times 2 matrix.
A second-hand bookstore sells textbooks at a markup of 50\%. The table shows the amounts they paid for old textbooks during the past academic year.
Organise the cost prices into a 5 \times 2 cost matrix.
Find the sales prices for each category and organise them into a 5 \times 2 revenue matrix.
Use \text{Profit} = \text{Revenue} - \text{Cost}, to construct the profit matrix.
How much profit will the bookstore generate in total if all stock is sold?
| Semester 1 | Semester 2 | |
|---|---|---|
| Business | \$940 | \$980 |
| Law | \$1020 | \$1170 |
| Mathematics | \$930 | \$1160 |
| Science | \$1180 | \$1040 |
| Engineering | \$1150 | \$970 |
A clothing store sells their clothes at a markup of 45\%. The table shows the amount they paid for the clothes in their summer and winter range:
Organise the cost prices into a 5 \times 2 cost matrix.
Find the sales prices for each category and organise them into a 5 \times 2 revenue matrix.
Use \text{Profit} = \text{Revenue} - \text{Cost}, to construct the profit matrix.
How much profit will the clothing store generate in total if all stock is sold?
| Summer | Winter | |
|---|---|---|
| Baby | \$500 | \$400 |
| Girls | \$540 | \$430 |
| Boys | \$450 | \$320 |
| Women | \$750 | \$690 |
| Men | \$760 | \$680 |
A bakery has three varieties of sausage rolls; meat, spinach, and mushroom. The bakery has two stores, which are both owned by the same person. One of the bakeries is near the beach and the other is in the city. The sales are split into morning and afternoon, and these sales are shown in the tables below:
Beach location:
| Meat | Spinach | Mushroom | |
|---|---|---|---|
| Morning | 10 | 9 | 6 |
| Afternoon | 18 | 12 | 7 |
City location:
| Meat | Spinach | Mushroom | |
|---|---|---|---|
| Morning | 22 | 16 | 20 |
| Afternoon | 38 | 32 | 29 |
Write the data from the Beach location into matrix B, where the rows are the times of day, and the columns are the types of sausage roll.
Write the data from the City location into matrix C, where the rows are the times of day, and the columns are the types of sausage roll.
Calculate the matrix sum B + C.
The profit on each sausage roll sold is \$2.80. Use matrix B + C to calculate the total profits.
A trading firm trades the futures and options markets, and employs both human traders and automated trading systems. The automated trading systems have proven to generate greater trading profits, but have an additional cost of improving and maintaining them. The first table shows the trading profits, in millions, generated by the firm's automated trading systems, while the second table shows the corresponding numbers by its human traders.
Trading profits by automated trading systems:
| 2011 | 2012 | 2013 | |
|---|---|---|---|
| Futures | 74 | 72 | 78 |
| Options | 70 | 69 | 65 |
Trading profits by human traders:
| 2011 | 2012 | 2013 | |
|---|---|---|---|
| Futures | 66 | 60 | 58 |
| Options | 62 | 65 | 59 |
Create a 2 \times 3 matrix that shows how much more trading profit the firm's automated trading systems generated in futures and options each year, compared to its human traders.
What was the overall percentage increase in profit from using automated trading systems in the futures market over human traders? Round your answer to one decimal place.
What was the overall percentage increase in profit from using automated trading systems in the options market over human traders? Round your answer to one decimal place.
The automated trading system needs to generate an excess trading profit of 20\% over the human traders in a market in order to cover the additional costs of maintaining and improving them. In which market would it be profitable for the firm to replace their human traders with an automated trading system, the futures market or the options market?
In an acrobatic gymnastics competition, each team's routine is given a score out of 30, with a maximum of 10 points each awarded for its Artistic component, Execution component and Difficulty component. Each team performs three routines, with the winner being the team with the highest points total at the end. The following tables show each team's scores for the Artistic, Execution and Difficulty components respectively for each of their routines:
Artistic Scores
| 1st Routine | 2nd Routine | 3rd Routine | |
|---|---|---|---|
| Team 1 | 8.3 | 7.9 | 7.4 |
| Team 2 | 9.2 | 8.4 | 8.0 |
| Team 3 | 7.8 | 7.5 | 6.8 |
Find the total scores for each team's routine. Express your answer as a 3 \times 3 matrix of the form:
\begin{matrix} & \begin{matrix} \text{1st} & \text{2nd} & \text{3rd} \end{matrix} \\ \begin{matrix} \text{Team 1} \\ \text{Team 2} \\ \text{Team 3} \end{matrix} & \begin{bmatrix} ⬚ & ⬚ & ⬚ \\ ⬚ & ⬚ & ⬚ \\ ⬚ & ⬚ & ⬚ \end{bmatrix} \end{matrix}Find each team's total score. Express your answer as a 3 \times 1 matrix.
Which team won the competition?
Execution Scores
| 1st Routine | 2nd Routine | 3rd Routine | |
|---|---|---|---|
| Team 1 | 7.3 | 8.9 | 6.4 |
| Team 2 | 8.2 | 9.0 | 8.9 |
| Team 3 | 6.8 | 6.9 | 7.2 |
Difficulty Scores
| 1st Routine | 2nd Routine | 3rd Routine | |
|---|---|---|---|
| Team 1 | 8.4 | 7.8 | 8.1 |
| Team 2 | 9.6 | 9.5 | 9.0 |
| Team 3 | 7.2 | 7.6 | 8.0 |
The following table provides some information about the people who went to the Mighty Imagination play centre today:
Organise this data into a 4 \times 2 matrix, M
What does m_{12} + m_{31} represent in this context?
What does m_{12} + m_{22} + m_{32} represent in this context?
Tomorrow is forecast to be raining for most of the day. The Mighty Imagination staff expect to see 20\% more people in each category as there was today.
| Male | Female | |
|---|---|---|
| Toddler | 32 | 24 |
| Child | 26 | 25 |
| Adolescent | 4 | 5 |
| Adult | 28 | 22 |
Multiply your matrix, M, to find the expected numbers of tomorrow's attendance at the play centre. Express your answer as a 4 \times 2 matrix, and round the elements to the nearest whole number.
The table shows the results for the top five AFL teams at the end of the most recent season. The winner is the team with the highest number of points at the end of the season. Wins are worth 4 points, draws are worth 2 points and losses are worth no points.
Organise the data into a 5 \times 3 matrix, A, where the rows are the teams and the columns are the wins, draws and losses.
Organise the number of points for a Win, Draw and Loss into a 3 \times 1 matrix, B.
Calculate each team's total points for the season by finding AB.
Which team won the competition?
| Wins | Draws | Losses | |
|---|---|---|---|
| Richmond | 16 | 5 | 1 |
| West Coast | 14 | 4 | 4 |
| Collingwood | 15 | 7 | 0 |
| Melbourne | 18 | 3 | 1 |
| Sydney | 13 | 7 | 2 |
The table shows the gross income of a married couple, Robert and Sarah, for the last year. The government charges income tax of 12\% for people earning \$35\,000 or more a year and 9\% for people earning less than \$35\,000 a year.
| Summer | Autumn | Winter | Spring | |
|---|---|---|---|---|
| Robert | \$9800 | \$8700 | \$8800 | \$8400 |
| Sarah | \$9900 | \$8600 | \$7800 | \$7200 |
Find Robert's annual income before tax.
Find Sarah's annual income before tax.
Find Robert and Sarah's tax rates and write them as decimals in a 1 \times 2 matrix, A.
Organise Robert and Sarah's incomes for each season into a 2 \times 4 matrix, B, with the rows representing Robert then Sarah, and the columns representing the seasons.
Find the matrix AB.
How much tax did the couple have to pay for their earnings in Winter and Spring?
Alexia and her family are heading off to dinner to celebrate her 12th birthday. Coming along with her are her parents, her two younger brothers and three younger sisters, and two of her grandparents. They are deciding between catching the bus or the train. The table below shows the various ticket prices for a return trip:
Organise the ticket prices into a 3 \times 2 matrix, A.
Create a 1 \times 3 matrix, B, that represents how many of each type of ticket (Child, Adult, Pensioner) needs to be bought.
Find BA.
Which mode of transport is the cheaper option?
| Bus | Train | |
|---|---|---|
| Child | 4.50 | 4.80 |
| Adult | 6.40 | 7.80 |
| Pensioner | 3.50 | 3.90 |
In a particular diving competition, each dive is scored by summing the scores given by the three judges and then multiplying this sum by the degree of difficulty of the dive. The first three tables show the scorecards of the three judges. The fourth table shows the degree of difficulty of the dive for each diver.
First Judge's Scorecard
| Dive 1 | Dive 2 | Dive 3 | |
|---|---|---|---|
| Caitlin | 9 | 6 | 5 |
| Ursula | 7 | 4 | 6.5 |
| Judy | 10 | 8 | 7 |
Third Judge's Scorecard
| Dive 1 | Dive 2 | Dive 3 | |
|---|---|---|---|
| Caitlin | 10 | 5.5 | 6 |
| Ursula | 8 | 4.5 | 6 |
| Judy | 8.5 | 7.5 | 6 |
Second Judge's Scorecard
| Dive 1 | Dive 2 | Dive 3 | |
|---|---|---|---|
| Caitlin | 8 | 5 | 4.5 |
| Ursula | 9 | 5 | 7 |
| Judy | 9.5 | 7 | 5.5 |
Degree of difficulty
| Caitlin | Ursula | Judy | |
|---|---|---|---|
| Dive 1 | 1.6 | 2.8 | 2.0 |
| Dive 2 | 2.3 | 2.2 | 1.8 |
| Dive 3 | 3.0 | 1.9 | 2.5 |
Find the 3 \times 3 matrix, A, which represents the sum of the judge's scores for each person's dive.
Create a 3 \times 3 matrix, D, representing the degree of difficulty of each dive. Where each row represents a dive, and each column represents a diver.
Find AD.
Which elements of AD contains each diver's final score for the competition? Explain your answer.
Which diver won the competition?
The table below shows three friends and their scores over four games played against each other. Let A and B be the matrices of ones as shown below:
| Game 1 | Game 2 | Game 3 | Game 4 | |
|---|---|---|---|---|
| Mandy | 10 | 11 | 16 | 20 |
| Millie | 9 | 15 | 12 | 19 |
| Max | 11 | 7 | 16 | 14 |
Orgainse the data from the table into a 3 \times 4 matrix, H.
Perform this matrix calculation.
Who had the highest average score?
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 A | Model B | Model C | Model D | Model E | |
|---|---|---|---|---|---|
| KC Hi Fi | 2 | 4 | 2 | 5 | 1 |
| Harley Morman | 1 | 2 | 3 | 4 | 2 |
| Good Gals | 6 | 0 | 4 | 3 | 2 |
| Wholesale price | \$90 | \$100 | \$110 | \$150 | \$190 |
| Retail Price | \$99 | \$120 | \$150 | \$220 | \$300 |
Create a 3 \times 5 matrix, S, to show the number of each coffee machine available at each store.
Create a 5 \times 2 matrix, T, to show the wholesale and retail prices of each coffee machine.
Find ST.
What does the matrix ST represent in this context?
Four different models of LED televisions are sold through two outlets. The stock on hand and corresponding wholesale and retail prices for each model and outlet are shown below:
| Model A | Model B | Model C | Model D | |
|---|---|---|---|---|
| KC Hi Fi | 4 | 3 | 0 | 3 |
| Good Gals | 7 | 4 | 1 | 2 |
| Wholesale price | \$90 | \$180 | \$300 | \$410 |
| Retail Price | \$169 | \$298 | \$598 | \$749 |
Find two matrices that can be multiplied to calculate the total wholesale and retail value of the TVs available at each store.
Multiply these two matrices.
If KC Hi Fi sells all of their LED TVs, how much profit will they make?
A new school year is approaching and Charlene is looking at the stationery lists of the items she needs to buy for her three children for school. Her oldest child Anthony, requires 10 pencils, 4 erasers, 12 markers and 4 highlighters. Isabel needs 8 pencils, 2 erasers, 8 markers and 4 highlighters and Reuben needs 2 of each of those items.
Create a 3 \times 4 matrix, A, that represents how many of each item the different children require.
Together with matrix A, consider the matrices of ones that are shown below:
B = \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}, C = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}, D = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, E = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}To calculate the total number of each item required, what matrix multiplication could be performed?
What would be the dimensions of the result of this matrix multiplication?
Perform this matrix multiplication.
Matrix A = \begin{bmatrix} 5 & 36 \end{bmatrix} shows the number of adults and then children attending a school trip to aquarium, and matrix B = \begin{bmatrix} 26 \\ 14 \end{bmatrix} shows the cost of an entry ticket for an adult and then a child. What matrix calculation will find the total cost of the group's entry?
Matrix A = \begin{bmatrix} 12 \\ 5 \\ 2 \end{bmatrix}, shows the price of fish, chips, and drinks at a local fish-and-chip shop. Matrix B = \begin{bmatrix} 23 & 31 & 33 \end{bmatrix}, shows the number of each that were ordered yesterday. What matrix calculation will find the total amount of money spent at the fish-and-chip shop?
The table below shows the number of people who visited a water park in 2018 and 2019 during each season:
| Summer | Autumn | Winter | Spring | |
|---|---|---|---|---|
| 2018 | 917 | 670 | 400 | 871 |
| 2019 | 990 | 760 | 629 | 800 |
Write the data from the table into a 2 \times 4 matrix. Let this matrix be A.
Together with matrix A, consider the matrices of ones that are shown below:
B = \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}, C = \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \end{bmatrix}, D = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, E = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix}To calculate the total number of people who visited the water park in each year, what matrix multiplication should be performed?
A movie complex offers the following three ticket packages:
The large group package in which you get 6 movie tickets, 6 drinks and 3 large boxes of popcorn.
The family package in which you get 4 movie tickets, 4 drinks and 2 large boxes of popcorn.
The couples package in which you get 2 movie tickets, no drinks and 1 large box of popcorn.
Represent this information in the 3 \times 3 matrix, M.
On a Saturday night, the movie complex sells 50 large group packages, 80 family packages and 100 couples packages. Write down the matrix that we need to multiply M by in order to determine the total number of movie tickets, drinks, and popcorn needed for all these packages.
Calculate the total number of tickets, drinks, and popcorn required for all of the Saturday night packages.
Jack and John are planning a fishing trip. Jack needs 2 sinkers, 3 packs of hooks and 4 jigs, while John needs 1 sinker, 4 packs of hooks and 2 jigs.
They can purchase the items at two local shops for the following prices:
- Fish Tackle: \$1.20 per sinker, \$4.00 for a pack of hooks, and \$2.20 per jig.
- Tackle Bite: \$1.90 per sinker, \$4.50 for a pack of hooks, and \$0.90 per jig.
Create a 3 \times 2 matrix, W, to show the required items for Jack and John.
Create a 2 \times 3 matrix, P, to show the prices for these items at the two stores.
To determine where Jack and John should shop should, what matrix multiplication should be performed?
Perform this matrix multiplication.
At which store should they each shop? Explain your answer.
A school's email system only allows students to send messages to their friends. As a result:
Jo can only email Bob and Ky.
-
Wes can only email Ky.
Bob can only email Jo and Ky.
Ky can only email Bob, Wes and Jo.
Create a matrix, A, that shows the number of direct paths of communication from each student to another student. For the rows and columns, use the order of students as: Wes then Bob then Jo then Ky.
Find A^{2}, which represents the two-step paths of communication between the students.
How many ways are there for Jo to email Ky through a third party?
The following map network shows connections between four towns:
Create the 4 \times 4 matrix M, to represent the direct one-step paths between the four locations. For the rows and columns, put the towns in alphabetical order.
Find the matrix M^{2}, which represents the two-step paths between the four locations.
The following map network shows the roads that connect three towns:
Create a 3 \times 3 matrix A, to represent the direct one-step paths between the three locations. For the rows and columns, use the order of Arwick, Bogville then Caraway.
Find A^{2}, which represents the two-step paths between the three locations.
How many two-step paths can be taken from Bogville to Carraway?
The following map network shows the roads that connect five towns:
Create a 5 \times 5 matrix M, to represent the direct one-step paths between the five locations. For the rows and columns, put the towns in alphabetical order.
Find M^{3}, which represents the three-step paths between the three locations.
How many three-step paths can be taken from Town D to Town E?
The map network below shows connections between five capital cities in Australia:
Create a 5 \times 5 matrix A, to represent the direct one-step paths between the five cities. For the rows and columns, use the order of Perth, Adelaide, Canberra, Sydney then Melbourne.
Find A^{2}, which represents the two-step paths between the three locations.
Are there any two cities that cannot be travelled between using a two-step path?