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Matrices, GEN2 2024 VCAA 9

Vince works on a construction site.

The amount Vince gets paid depends on the type of shift he works, as shown in the table below.

\begin{array}{|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \textbf{Shift type} \rule[-1ex]{0pt}{0pt}& \textbf{Normal} & \textbf{Overtime} & \textbf {Weekend} \\
\hline
\rule{0pt}{2.5ex} \textbf{Hourly rate of pay} \rule[-1ex]{0pt}{0pt} \ \text{(\$ per hour)} & 36 & 54 & 72 \\
\hline
\end{array}

This information is shown in matrix \(R\) below.

\begin{align*}
R=\left[\begin{array}{lll}
36 & 54 & 72
\end{array}\right] \end{align*}

  1. Matrix \(R^T\) is the transpose of matrix \(R\).
  2. Determine the matrix \(R^T\).   (1 mark)

During one week, Vince works 28 hours at the normal rate of pay, 6 hours at the overtime rate of pay, and 8 hours at the weekend rate of pay.

  1. Complete the following matrix calculation showing the total amount Vince has been paid for this week.  (1 mark)

  

Vince will receive $90 per hour if he works a public holiday shift.

Matrix \(Q\), as calculated below, can be used to show Vince's hourly rate for each type of shift.

\begin{align*}
\begin{aligned}
Q & =n \times\left[\begin{array}{llll}
1 & 1.5 & 2 & p
\end{array}\right] \\
& =\left[\begin{array}{llll}
36 & 54 & 72 & 90
\end{array}\right] \end{aligned}
\end{align*}

  1. Write the values of \(n\) and \(p\).  (1 mark)

Show Answers Only

a.   \(R^T=\begin{bmatrix}
36 \\
54 \\
72
\end{bmatrix}\)

b.    \([28\quad  6\quad  8]\times R^T = [1908]\)

c.    \(n=36\ ,\ p=2.5\)

Show Worked Solution

a.   \(R^T=\begin{bmatrix}
36 \\
54 \\
72
\end{bmatrix}\)

 
b.    \(\begin{bmatrix}
28 & 6 & 8
\end{bmatrix}\times\ R^T=\begin{bmatrix}
28\times36 + 6\times54+ 8\times 72
\end{bmatrix}=[1908]\)

 
c.   \(n=\ \text{Normal hourly rate}\ =36\)

\(p=\ \text{Overtime rate}\ =\dfrac{90}{36}=2.5\)

MATRICES, FUR2 2020 VCAA 1

The three major shopping centres in a large city, Eastmall `(E)`, Grandmall `(G)` and Westmall `(W)`, are owned by the same company.

The total number of shoppers at each of the centres at 1.00 pm on a typical day is shown in matrix `V`.

`qquad qquad qquad {:(qquad qquad qquad \ E qquad qquad G qquad qquad \  W),(V = [(2300,2700,2200)]):}`

  1. Write down the order of matrix `V`.   (1 mark)

Each of these centres has three major shopping areas: food `(F)`, clothing `(C)` and merchandise `(M)`.

The proportion of shoppers in each of these three areas at 1.00 pm on a typical day is the same at all three centres and is given in matrix `P` below

`qquad qquad qquad P = [(0.48), (0.27), (0.25)] {:(F),(C),(M):}

  1. Grandmall’s management would like to see 700 shoppers in its merchandise area at 1.00 pm.

     

    If this were to happen, how many shoppers, in total, would be at Grandmall at this time?   (1 mark)

  2. The matrix  `Q = P xx V`  is shown below. Two of the elements of this matrix are missing.
     
    `{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, \ text{___}, 1056 ), (621,\ text{___}, 594), (575, 675, 550)]{:(F),(C), (M):}):}`
     
    1. Complete matrix `Q` above by filling in the missing elements.   (1 mark)

    2. The element in row `i` and column `j` of matrix `Q` is `q_(ij)`.
    3. What does the element `q_23` represent?   (1 mark)

The average daily amount spent, in dollars, by each shopper in each of the three areas at Grandmall in 2019 is shown in matrix  `A_2019`  below.

`qquad qquad A_2019 = [(21.30), (34.00), (14.70)] {:(F),(C),(M):}`

On one particular day, 135 shoppers spent the average daily amount on food, 143 shoppers spent the average daily amount on clothing and 131 shoppers spent the average daily amount on merchandise.

  1. Write a matrix calculation, using matrix  `A_2019`, showing that the total amount spent by all these shoppers is $9663.20   (1 mark)

  2. In 2020, the average daily amount spent by each shopper was expected to change by the percentage shown in the table below.
      

     

      Area food clothing merchandise
      Expected change     increase by 5%       decrease by 15%       decrease by 1%   

     

     

    The average daily amount, in dollars, expected to be spent in each area in 2020 can be determined by forming the matrix product

  3. `qquad qquad A_2020 = K xx A_2019`
  4. Write down matrix `K`.    (1 mark)

Show Answers Only
  1. `1 xx 3`
  2. `2800`
  3.  i.   `{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, 1296, 1056 ), (621, 729, 594), (575, 675, 550)]{:(F),(C), (M):}):}`
     
  4. ii. `q_23\ text(represents the number of people)`
  5.    `text(in the clothing area of Westmall.)`
  6. `text(Total spent) = [(135, 143, 131)][(21.30), (34.00), (14.70)] = [9663.20]`
  7. `K = [(1.05, 0, 0),(0, 0.85, 0),(0, 0, 0.99)]`
Show Worked Solution

a.  `1 xx 3`

♦ Mean mark part (b) 45%.
b.   `0.25 xx G\ text(shoppers in)\ M` `= 700`
  `:. G\ text(shoppers in)\ M` `= 700/0.25`
    `= 2800`

 

c.i.   `{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, 1296, 1056 ), (621, 729, 594), (575, 675, 550)]{:(F),(C), (M):}):}`

 

c.ii.   `q_23\ text(represents the number of people)`
  `text(in the clothing area of Westmall.)`

 

d.  `text(Total spent) = [(135, 143, 131)] [(21.30), (34.00), (14.70)] = [9663.20]`

♦♦ Mean mark part (e) 24%.
 

e.   `A_2020` `= K xx [(21.30), (34.00), (14.70)]`
  `:. K` `= [(1.05, 0, 0),(0, 0.85, 0),(0, 0, 0.99)]`

MATRICES, FUR2-NHT 2019 VCAA 1

A total of six residents from two towns will be competing at the International Games.

Matrix `A`, shown below, contains the number of male `(M)` and the number of female `(F)` athletes competing from the towns of Gillen `(G)` and Haldaw `(H)`.

`{:(qquad qquad quad \ M quad F), (A = [(2, 2), (1, 1)]{:(G),(H):}):}` 

  1. How many of these athletes are residents of Haldaw?   (1 mark)

Each of the six athletes will compete in one event: table tennis, running or basketball.

Matrices `T` and `R`, shown below, contain the number of male and female athletes from each town who will compete in table tennis and running respectively.
 

            Table tennis                        Running             
 

`{:(qquad qquad quad \ M quad F), (T = [(0, 1), (1, 0)]{:(G),(H):}):}`

`{:(qquad qquad quad \ M quad F), (R = [(1, 1), (0, 0)]{:(G),(H):}):}`

 

  1. Matrix `B` contains the number of male and female athletes from each town who will compete in basketball.

     

    Complete matrix `B` below.   (1 mark)

`{:(qquad qquad qquad \ M qquad quad F), (B = [(\ text{___}, text{___}\ ), (\ text{___}, text{___}\)]{:(G),(H):}):}`

Matrix `C` contains the cost of one uniform, in dollars, for each of the three events: table tennis `(T)`, running `(R)` and basketball `(B)`.

`C = [(515), (550), (580)]{:(T), (R), (B):}`

    1. For which event will the total cost of uniforms for the athletes be $1030?   (1 mark)

    2. Write a matrix calculation, that includes matrix `C`, to show that the total cost of uniforms for the event named in part c.i. is contained in the matrix answer of [1030].   (1 mark)

  1. Matrix `V` and matrix `Q` are two new matrices where  `V = Q xx C`  and:
  • matrix `Q` is a  `4 xx 3`  matrix
  • element `v_11 =` total cost of uniforms for all female athletes from Gillen
  • element `v_21 =` total cost of uniforms for all female athletes from Haldaw
  • element `v_31 =` total cost of uniforms for all male athletes from Gillen
  • element `v_41 =` total cost of uniforms for all male athletes from Haldaw
     
  • `C = [(515), (550), (580)]{:(T), (R), (B):}`
  1. Complete matrix `Q` with the missing values.   (1 mark)

`Q = [(1, text{___}, text{___}\ ), (0, 0, 1), (0, 1, 1), (\ text{___}, text{___}, 0)]`

Show Answers Only
  1.  `2`
  2.  `B = [(1, 0), (0, 1)]`
  3. i.  `text(Table tennis)`
    ii. `[2\ \ \ 0\ \ \ 0] xx [(515), (550), (580)] = [1030]`
  4.  `Q = [(1, \ 1, \ 0),(0, \ 0, \ 1),(0, \ 1, \ 1),(1, \ 0, \ 0)]`
Show Worked Solution

a.  `2`
 

b.  `B = [(1, 0), (0, 1)]`
 

c.i.  `text(Table tennis)`
 

c.ii.  `[2\ \ \ 0\ \ \ 0] xx [(515), (550), (580)] = [1030]`
 

d.  `Q = [(1, \ 1, \ 0),(0, \ 0, \ 1),(0, \ 1, \ 1),(1, \ 0, \ 0)]`

MATRICES, FUR2 2019 VCAA 1

The car park at a theme park has three areas, `A, B` and `C`.

The number of empty `(E)` and full `(F)` parking spaces in each of the three areas at 1 pm on Friday are shown in matrix `Q`  below.
 

`{:(qquad qquad qquad \ E qquad F),(Q = [(70, 50),(30, 20),(40, 40)]{:(A),(B),(C):}quad text(area)):}`
 

  1. What is the order of matrix `Q`?   (1 mark)

  2. Write down a calculation to show that 110 parking spaces are full at 1 pm.   (1 mark)

Drivers must pay a parking fee for each hour of parking.

Matrix `P`, below, shows the hourly fee, in dollars, for a car parked in each of the three areas.
 

`{:(qquad qquad qquad qquad qquad text{area}), (qquad qquad qquad A qquad quad quad B qquad qquad C), (P = [(1.30, 3.50, 1.80)]):}`
 

  1. The total parking fee, in dollars, collected from these 110 parked cars if they were parked for one hour is calculated as follows.  

     

     

    `qquad qquad qquad P xx L = [207.00]`

     

    where matrix  `L`  is a  `3 xx 1`  matrix.

     

    Write down matrix  `L`.   (1 mark)

The number of whole hours that each of the 110 cars had been parked was recorded at 1 pm. Matrix `R`, below, shows the number of cars parked for one, two, three or four hours in each of the areas `A, B` and `C`.

`{:(qquadqquadqquadqquadquadtext(area)),(quad qquadqquadquad \ A qquad B qquad C),(R = [(3, 1, 1),(6, 10, 3),(22, 7,10),(19, 2, 26)]{:(1),(2),(3),(4):}\ text(hours)):}`
 

  1. Matrix  `R^T`  is the transpose of matrix  `R`.

      

    Complete the matrix  `R^T`  below.   (1 mark)

      

    `qquad R^T = [( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , )]`
     

  2. Explain what the element in row 3, column 2 of matrix  `R^T`  represents.   (1 mark)

Show Answers Only
  1. `3 xx 2`
  2. `50 + 20 + 40 = 110`
  3. `L = [(50), (20), (40)]`
  4. `R^T = [(3 ,6 , 22, 19), (1, 10, 7, 2), (1, 3, 10, 26)]`
  5. `text(Number of cars parked in area)\ C\ text(for 2 hours).`
Show Worked Solution

a.  `text(Order) : 3 xx 2`
 

b.  `text(Add 2nd column): \ 50 + 20 + 40 = 110`
 

c.  `L = [(50), (20), (40)]`
 

d.  `R^T = [(3 ,6 , 22, 19), (1, 10, 7, 2), (1, 3, 10, 26)]`
 

e.   `e_32\ text(in)\ R^T =>` `text(number of cars parked in area)\ C`
    `text(for 2 hours.)`

MATRICES, FUR2 2007 VCAA 1

The table below displays the energy content and amounts of fat, carbohydrate and protein contained in a serve of four foods: bread, margarine, peanut butter and honey.
 

MATRICES, FUR2 2007 VCAA 1
 

  1. Write down a 2 x 3 matrix that displays the fat, carbohydrate and protein content (in columns) of bread and margarine.   (1 mark)

  2. `A` and `B` are two matrices defined as follows.
     
         `A = [(2,2,1,1)]`     `B = [(531),(41),(534),(212)]`

    1. Evaluate the matrix product  `AB`.   (1 mark)

    2. Determine the order of matrix product  `BA`.   (1 mark)

Matrix `A` displays the number of servings of the four foods: bread, margarine, peanut butter and honey, needed to make a peanut butter and honey sandwich.

Matrix `B` displays the energy content per serving of the four foods: bread, margarine, peanut butter and honey.

    1. Explain the information that the matrix product `AB` provides.   (1 mark)

  1. The number of serves of bread (`b`), margarine (`m`), peanut butter (`p`) and honey (`h`) that contain, in total, 53 grams of fat, 101.5 grams of carbohydrate, 28.5 grams of protein and 3568 kilojoules of energy can be determined by solving the matrix equation
      
         `[(1.2,6.7,10.7,0),(20.1,0.4,3.5,12.5),(4.2,0.6,4.6,0.1),(531,41,534,212)][(b),(m),(p),(h)] = [(53),(101.5),(28.5),(3568)]`
      
    Solve the matrix equation to find the values `b`, `m`, `p` and `h`.   (2 marks)

Show Answers Only
  1.  
    `[(1.2,20.1,4.2),(6.7,0.4,0.6)]`
    1. `[1890]`
    2. `underset (4 xx 1) B xx underset (1 xx 4) A = underset (4 xx 4) (BA)`
    3. `BA\ text(provides the total energy)`
      `text(content of the servings of these)`
      `text(four foods in one sandwich.)`
  3. `b = 4, m = 4, p = 2, h = 1`
Show Worked Solution
a.    `[(1.2,20.1,4.2),(6.7,0.4,0.6)]`

 

b.i.    `AB` `= [(2, 2, 1, 1)] [(531), (41), (534), (212)]`
    `= [1890]`

 

b.ii.   `underset (4 xx 1) B xx underset (1 xx 4) A = underset (4 xx 4) (BA)`

 

b.iii.   `BA\ text(provides the total energy content of the)`
 

`text(servings of these four foods in one sandwich.)`

 

c.    `[(b),(m),(p),(h)]` `= [(1.2,6.7,10.7,0),(20.1,0.4,3.5,12.5),(4.2,0.6,4.6,0.1),(531,41,534,212)]^(-1)[(53),(101.5),(28.5),(3568)]`
    `= [(4),(4),(2),(1)]\ \ \ text{(by graphics calculator)}`

 
`:. b = 4, m = 4, p = 2\ text(and)\ h = 1.`

MATRICES, FUR2 2010 VCAA 1

In a game of basketball, a successful shot for goal scores one point, two points, or three points, depending on the position from which the shot is thrown.

`G`  is a column matrix that lists the number of points scored for each type of successful shot.

`G = [(1),(2),(3)]`

In one game, Oscar was successful with

    • 4 one-point shots for goal
    • 8 two-point shots for goal
    • 2 three-point shots for goal.
  1. Write a row matrix, `N`, that shows the number of each type of successful shot for goal that Oscar had in that game.   (1 mark)

  2. Matrix `P` is found by multiplying matrix `N` with matrix `G` so that  `P = N xx G`
  3. Evaluate matrix `P`.   (1 mark)

  4. In this context, what does the information in matrix `P` provide?   (1 mark)

Show Answers Only
  1. `N = [(4, 8, 2)]`
  2. `P = [26]`
  3. `text(The total points scored by Oscar in the game.)`
Show Worked Solution

a.   `N = [(4, 8, 2)]`
 

b.    `P` `= NG`
    `= [(4, 8, 2)][(1),(2),(3)]`
    `= [26]`

 
c.   `text(The total points scored by Oscar in the game.)`

MATRICES, FUR2 2011 VCAA 1

The diagram below shows the feeding paths for insects (`I`), birds (`B`) and lizards (`L`). The matrix `E` has been constructed to represent the information in this diagram. In matrix `E`, a 1 is read as "eat" and a  0  is read as "do not eat".
 

MATRICES, FUR2 2011 VCAA 11

  1. Referring to insects, birds or lizards
  2.  i. what does the 1 in column `B`, row `L`, of matrix `E` indicate?   (1 mark)

  3. ii. what does the row of zeros in matrix `E` indicate?   (1 mark)

The diagram below shows the feeding paths for insects (`I`), birds (`B`), lizards (`L`) and frogs (`F`).

The matrix `Z` has been set up to represent the information in this diagram.

Matrix `Z` has not been completed.
 

MATRICES, FUR2 2011 VCAA 12

  1. Complete the matrix `Z` above by writing in the seven missing elements.   (1 mark)

Show Answers Only

a.i.    `text(Birds eat lizards)`

a.ii.   `text(Insects, birds or lizards do not eat birds)`

b.
      `{:((qquadqquadquadI,B,L,F)),(Z = [(0,1,1,1),(0,0,0,0),(0,1,0,0),(0,1,1,0)]{:(I),(B),(L),(F):}):}`

Show Worked Solution

a.i.   `text(The 1 represents that birds eat lizards.)`
 

a.ii.   `text(It indicates that insects, birds or lizards DO NOT)`

   `text(eat birds.)`
 

b.    `{:((qquadqquadquadI,B,L,F)),(Z = [(0,1,1,1),(0,0,0,0),(0,1,0,0),(0,1,1,0)]{:(I),(B),(L),(F):}):}`

MATRICES, FUR2 2013 VCAA 1

Five trout-breeding ponds, `P`, `Q`, `R`, `X` and `V`, are connected by pipes, as shown in the diagram below.
 

Matrices, FUR2 2013 VCAA 1 

The matrix `W` is used to represent the information in this diagram.

`{:({:\ qquadqquadqquadPquadQquad\ Rquad\ Xquad\ V:}),(W = [(0,1,1,1,0), (1,0,0,1,0),(1,0,0,1,0),(1,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

In matrix `W`

•  the 1 in column 1, row 2, for example, indicates that a pipe directly connects pond `P` and pond `Q`

•  the 0 in column 1, row 5, for example, indicates that pond `P` and pond `V` are not directly connected by a pipe.

  1. Find the sum of the elements in row 3 of matrix `W`.   (1 mark)

  2. In terms of the breeding ponds described, what does the sum of the elements in row 3 of matrix `W` represent?   (1 mark)

The pipes connecting pond `P` to pond `R` and pond `P` to pond `X` are removed.

Matrix `N` will be used to show this situation. However, it has missing elements.

  1. Complete matrix `N` below by filling in the missing elements in row 1 and column 1.   (1 mark)

        
             Matrices, FUR2 2013 VCAA 1_c

Show Answers Only
  1. `2`
  2. `text(The sum means that 2 other)`
    `text(ponds connect directly to pond)\ R.`
  3.  
    `{:({:qquadqquadqquadPquadQquadRquadXquadV:}),(N = [(0,1,0,0,0),(1,0,0,1,0),(0,0,0,1,0),(0,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`
Show Worked Solution

a.   `1+ 0 + 0 +1+ 0 = 2`
 

b.   `text(The sum means that 2 other ponds)`

`text(connect directly to pond)\ R.`
 

c.    `{:({:qquadqquadqquadPquadQquad\ Rquad\ Xquad\ V:}),(N = [(0,1,0,0,0),(1,0,0,1,0),(0,0,0,1,0),(0,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

MATRICES, FUR1 2009 VCAA 3 MC

The number of people attending the morning, afternoon and evening sessions at a cinema is given in the table
below. The admission charges (in dollars) for each session are also shown in the table.
 

     MATRICES, FUR1 2009 VCAA 3 MC 1
 

A column matrix that can be used to list the number of people attending each of the three sessions is

A.   `[25,56,124]`

B.   `[[25],[56],[124]]`

C.   `[12,15,20]`

D.   `[[12],[15],[20]]`

E.   `[[25,56,124],[12,15,20]]`

 

Show Answers Only

`B`

Show Worked Solution

`text(S)text(ince the answer must be a “column” matrix)`

`text(of the attendees.)` 

`=>  B`

MATRICES, FUR1 2012 VCAA 4 MC

The diagram below shows the tracks directly linking four camping sites `P, Q, R` and `S` in a national park.

The shortest time that it takes to walk between the camping sites (in minutes), along each of these tracks, is also shown.
 

MATRICES, FUR1 2012 VCAA 4 MC

 
A matrix that could be used to present the same information is

MATRICES, FUR1 2012 VCAA 4 MC ab

MATRICES, FUR1 2012 VCAA 4 MC cd

MATRICES, FUR1 2012 VCAA 4 MC e

Show Answers Only

`E`

Show Worked Solution

`text(Presenting the “same information” involves)`

`text(recording the exact walk times in the matrix.)`

`rArr E`

MATRICES, FUR1 2013 VCAA 5 MC

Five students, Richard (R), Brendon (B), Lee (L), Arif (A) and Karl (K), were asked whether they played each of the following sports, football (F), golf (G), soccer (S) or tennis (T). Their responses are displayed in the table below.
 

MATRICES, FUR1 2013 VCAA 5 MC
 

If 1 is used to indicate that the student plays a particular sport and 0 is used to indicate that the student does not play a particular sport, which one of the following matrices could be used to represent the information in the table?
 

MATRICES, FUR1 2013 VCAA 5 MC ab1

MATRICES, FUR1 2013 VCAA 5 MC cd1

MATRICES, FUR1 2013 VCAA 5 MC e

Show Answers Only

`C`

Show Worked Solution

`rArr C`

MATRICES, FUR1 2006 VCAA 4 MC

Three teams, Blue (`B`), Green (`G`) and Red (`R`), compete for three different sporting competitions.

The table shows the competition winners for the past three years.
 

MATRICES, FUR1 2006 VCAA 4 MC
 

A matrix that shows the total number of competitions won by each of the three teams in each of these three years could be
 

MATRICES, FUR1 2006 VCAA 4 MC ab

MATRICES, FUR1 2006 VCAA 4 MC cd

MATRICES, FUR1 2006 VCAA 4 MC e

Show Answers Only

`B`

Show Worked Solution

`rArr B`