If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka



Word Problems on Matrices

Samuel Dominic Chukwuemeka (SamDom For Peace) You may verify your answers as applicable with: Matrices Calculators
For ACT Students
The ACT is a timed exam...$60$ questions for $60$ minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no "negative" penalty for any wrong answer.

For JAMB and CMAT Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.

Solve these matrix applications (word problems).
Show all work.
Indicate the method used to solve each problem as applicable.
Some of the questions have hints. The hint indicates what matrix operation you should use.

(1.) Arizona Western College (AWC) Airlines (Matadors Airlines) operates three flights one-way on any given day between San Luis, AZ; and New York City.
There are three cabin classes in each flight. The cabin classes are First Class, Business Class, and Economy Class.
Flight $1$ has $30$ First Class seats, $60$ Business Class seats, and $120$ Economy Class seats.
Flight $2$ has $40$ First Class seats, $90$ Business Class seats, and $100$ Economy Class seats.
Flight $3$ has $50$ First Class seats, $70$ Business Class seats, and $160$ Economy Class seats.
The fares per passenger seat for each cabin class are:
First Class = $$2000.00$ per seat
Business Class = $$1500.00$ per seat
Economy Class = $$1000.00$ per seat
Assume all seats are filled on a specific day, say $Day\: 3$; calculate the revenues generated by each of the flights on $Day\: 3$.
Hint: Matrix Multiplication


Let FC = First Class
BC = Business Class
EC = Economy Class

First Step: Represent the information about the cabin classes and the number of seats in each cabin as a $3 * 3$ matrix.
Cabins (FC, BC, EC)

$ Flights \begin{bmatrix} 30 & 60 & 120 \\[2ex] 40 & 90 & 100 \\[2ex] 50 & 70 & 160 \end{bmatrix} $

Let Matrix A = $ \begin{bmatrix} 30 & 60 & 120 \\[2ex] 40 & 90 & 100 \\[2ex] 50 & 70 & 160 \end{bmatrix} $

Second Step: Represent the informatoon about the fares/charges per passenger seat as a $3 * 1$ matrix.

Let Matrix B = $ \begin{bmatrix} 2000 \\[2ex] 1500 \\[2ex] 1000 \end{bmatrix} $

Third Step: Determine the revenue from each flight by multiplying Matrix $A$ by Matrix $B$
In other words, the number of seats per flight multiplied by the charge per seat gives the revenue.

$ \begin{bmatrix} 30 & 60 & 120 \\[2ex] 40 & 90 & 100 \\[2ex] 50 & 70 & 160 \end{bmatrix} $ * $ \begin{bmatrix} 2000 \\[2ex] 1500 \\[2ex] 1000 \end{bmatrix} $ = $ AB $

Order of $A = 3 * 3$
Order of $B = 3 * 1$
$A$ and $B$ can multiply.
Order of $AB = 3 * 1$

$ 30(2000) + 60(1500) + 120(1000) = 60000 + 90000 + 120000 = 270000 \\[2ex] 40(2000) + 90(1500) + 100(1000) = 80000 + 135000 + 100000 = 315000 \\[2ex] 50(2000) + 70(1500) + 160(1000) = 100000 + 105000 + 160000 = 365000 $

$ AB = \begin{bmatrix} 270000 \\[2ex] 315000 \\[2ex] 365000 \end{bmatrix} $

Interpretation: This means that on $Day\: 3$;
Flight $1$ generated a revenue of $$270,000.00$
Flight $2$ generated a revenue of $$315,000.00$
Flight $3$ generated a revenue of $$365,000.00$
(2.) ACT Valley High School and Mountain High School have decided that selected students will attend a daytime theatrical performance that costs $\$5$ for each teacher and $\$3$ for each student.
One teacher and $10$ students from Valley High will attend, and $2$ teachers and $25$ students from Mountain High will attend.
Which of the following matrix products represents the ticket costs, in dollars, for each high school?

$ A.\:\: \begin{bmatrix} 5 & 3 \end{bmatrix}\begin{bmatrix} 1 & 2 \\[2ex] 10 & 25 \end{bmatrix} \\[5ex] B.\:\: \begin{bmatrix} 5 & 3 \end{bmatrix}\begin{bmatrix} 1 & 10 \\[2ex] 25 & 2 \end{bmatrix} \\[5ex] C.\:\: \begin{bmatrix} 5 & 3 \end{bmatrix}\begin{bmatrix} 1 & 25 \\[2ex] 2 & 10 \end{bmatrix} \\[5ex] D.\:\: \begin{bmatrix} 5 \\[2ex] 3 \end{bmatrix}\begin{bmatrix} 1 & 2 \\[2ex] 10 & 25 \end{bmatrix} \\[5ex] E.\:\: \begin{bmatrix} 5 \\[2ex] 3 \end{bmatrix}\begin{bmatrix} 1 & 10 \\[2ex] 2 & 25 \end{bmatrix} \\[3ex] $

Try to solve without Matrices

Valley High School
1 teacher
10 students
Cost = $5 * 1 + 3 * 10$

Mountain High School
2 teacher
25 students
Cost = $5 * 2 + 3 * 25$

We expect to have two answers: one for each school
The two answers represents either a $1 * 2$ matrix or a $2 * 1$ matrix
If the answer is a $2 * 1$ matrix, then we need to multiply a $2 * 2$ matrix and a $2 * 1$ matrix...Multiplication of Matrices
That option is not included.
So, we need a $1 * 2$ matrix
To have a $1 * 2$ answer, we need to multiply a $1 * 2$ matrix and a $2 * 2$ matrix ... Multiplication of Matrices
This means that we are left with Options $A$, $B$, and $C$
Observing the trend/sequence based on our calculation, the product matrix will be:
$ \begin{bmatrix} 5 & 3 \end{bmatrix}\begin{bmatrix} 1 & 2 \\[2ex] 10 & 25 \end{bmatrix} $
(3.) CSEC In a football tournament, points are awarded as follows: $3$ points for a win, $1$ point for a draw and $0$ points for a loss.
(i) Write a $3 * 1$ matrix, $P$, to represent this information.

During the tournament, Team Alpha recorded $5$ wins, $1$ draw and $3$ losses, while Team Beta recorded $3$ wins, $4$ draws, and $2$ losses.
(ii) Write a $2 * 3$ matrix, $R$ to represent this information.
(iii) Calculate the matrix product $RP$
(iv) What does the matrix product $RP$ represent?


Row by Column....Multiplication of Matrices

$ \begin{bmatrix} 5 & 1 & 3 \rightarrow Team\:A \\[2ex] 3 & 4 & 2 \rightarrow Team\:B \end{bmatrix} \begin{bmatrix} 3 \rightarrow Win \\[2ex] 1 \rightarrow Draw \\[2ex] 0 \rightarrow Loss \end{bmatrix} \\[5ex] (i)\:\: P = \begin{bmatrix} 3 \\[2ex] 1 \\[2ex] 0 \end{bmatrix} \\[7ex] (ii)\:\: R = \begin{bmatrix} 5 & 1 & 3 \\[2ex] 3 & 4 & 2 \\[2ex] \end{bmatrix} \\[7ex] RP = \begin{bmatrix} 5 & 1 & 3 \\[2ex] 3 & 4 & 2 \end{bmatrix} \begin{bmatrix} 3 \\[2ex] 1 \\[2ex] 0 \end{bmatrix} \\[7ex] R * P = RP \\[3ex] Order: 2\:by\:3 * 3\:by\:1 \rightarrow 2\:by\:1 \\[3ex] (iii)\:\: RP = \begin{bmatrix} 5(3) + 1(1) + 3(0) \\[2ex] 3(3) + 4(1) + 2(0) \end{bmatrix} = \begin{bmatrix} 15 + 1 + 0 \\[2ex] 9 + 4 + 0 \end{bmatrix} = \begin{bmatrix} 16 \\[2ex] 13 \end{bmatrix} \\[3ex] $ (iv) The matrix product $RP$ represents the total number of points for Team Alpha and Team Beta in a football tournament.
Team Alpha had $16$ points
Team Beta had $13$ points
(4.)

(5.) ACT A $500-square-mile$ national park in Kenya has large and small protected animals.
The number of large protected animals at the beginning of $201$ is given in the table below.
Large animal Number
Elephant
Rhinoceros
Lion
Leopard
Zebra
Giraffe
$600$
$100$
$200$
$300$
$400$
$800$
Total $2,400$

At the beginning of $2014$, the number of all protected animals in the park was $10,000$
Zoologists predict that for each year from $2015$ to $2019$, the total number of protected animals in the park at the beginning of the year will be $2\%$ more than the number of protected animals in the park at the beginning of the previous year.

In this park, the average number of gallons of water consumed per day by each elephant, lion, and giraffe is $50$, $5$, and $10$, respectively.
Which of the following matrix products yields the average total number of gallons of water consumed per day by all the elephants, lions, and giraffes in the park?

$ A.\:\: \begin{bmatrix} 600 & 200 & 800 \end{bmatrix}\begin{bmatrix} 50 \\[2ex] 5 \\[2ex] 10 \end{bmatrix} \\[5ex] B.\:\: \begin{bmatrix} 600 & 800 & 200 \end{bmatrix}\begin{bmatrix} 50 \\[2ex] 5 \\[2ex] 10 \end{bmatrix} \\[5ex] C.\:\: \begin{bmatrix} 600 \\[2ex] 200 \\[2ex] 800 \end{bmatrix}\begin{bmatrix} 50 & 5 & 10 \end{bmatrix} \\[5ex] D.\:\: \begin{bmatrix} 600 \\[2ex] 800 \\[2ex] 200 \end{bmatrix}\begin{bmatrix} 50 & 5 & 10 \end{bmatrix} \\[5ex] E.\:\: \begin{bmatrix} 600 \\[2ex] 800 \\[2ex] 200 \end{bmatrix}\begin{bmatrix} 50 \\[2ex] 5 \\[2ex] 10 \end{bmatrix} \\[5ex] $

If you don't know how to begin, try to solve it without Matrices.

$ Average\;\;total\;\;number\;\;of\;\;gallons \\[3ex] = Average\;\;number\;\;of\;\;gallons\;\;consumed\;\;by\;\;each\;\;elephant * Number\;\;of\;\;elephants \\[3ex] + \\[3ex] Average\;\;number\;\;of\;\;gallons\;\;consumed\;\;by\;\;each\;\;lion * Number\;\;of\;\;lions \\[3ex] + \\[3ex] Average\;\;number\;\;of\;\;gallons\;\;consumed\;\;by\;\;each\;\;giraffe * Number\;\;of\;\;giraffes \\[3ex] = 50(600) + 5(200) + 10(800) \\[3ex] $ We expect to have only one answer.
Bring it on to Matrices:
One answer means a $1 * 1$ matrix
Go back to the multiplication:
We are dealing with three animals: elephant, lion, and giraffe
To obtain a $1 * 1$ matrix as the product, we will need a $1 * 3$ matrix to multiply a $3 * 1$ ... Multiplication of Matrices
So, we eliminate Options $C$, $D$, and $E$
We are left with Options $A$ and $B$
The multiplication we did above is also the same as writing it this way

$ = 600(50) + 200(5) + 800(10) \\[3ex] $ The correct option is $A$
(6.)


$ \begin{vmatrix} p & c \\ -d & -e \end{vmatrix} \\[3ex] = (p)(-e) - (-d)(c) \\[2ex] = -ep - (-cd) \\[2ex] = -ep + cd \\[2ex] = cd - ep $