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Matrices, GEN1 2024 VCAA 28 MC

A primary school is hosting a sports day.

Students represent one of four teams: blue \((B)\), green \((G)\), red \((R)\) or yellow \((Y)\).

Students compete in one of three sports: football \((F)\), netball \((N)\) or tennis \((T)\).

Matrix \(W\) shows the number of students competing in each sport and the team they represent.

\begin{aligned} \\
& \quad B  \quad \ G \quad \  R \quad \ Y \\
W = & \begin{bmatrix}
85 & 60 & 64 & 71 \\
62 & 74 & 80 & 64 \\
63 & 76 & 66 & 75
\end{bmatrix}\begin{array}{l}
F\\
N\\
T
\end{array}
\end{aligned}

Matrix \(W\) is multiplied by the matrix \(\begin{bmatrix}1 \\ 1 \\ 1 \\ 1\end{bmatrix}\) to produce matrix \(X\).

Element \(x_{31}\) indicates that

  1. 210 students represent the blue team.
  2. 210 students compete in netball.
  3. 280 students compete in tennis.
  4. 280 students compete in football.
Show Answers Only

\(C\)

Show Worked Solution

\begin{aligned} \\
X = & \begin{bmatrix}
280 \\
280 \\
280
\end{bmatrix}\begin{array}{l}
F\\
N\\
T
\end{array}
\end{aligned}

\(\Rightarrow C\)

Matrices, GEN2 2023 VCAA 8

A circus sells three different types of tickets: family \((F)\), adult \((A)\) and child \((C)\).

The cost of admission, in dollars, for each ticket type is presented in matrix \(N\) below.

\(N=\begin{bmatrix}
36 \\
15 \\
8
\end{bmatrix}\begin{aligned}
F \\
A \\
C
\end{aligned}\)

The element in row \(i\) and column \(j\) of matrix \(N\) is \(n_{i j}\).

  1. Which element shows the cost for one child ticket?  (1 mark)

  2. A family ticket will allow admission for two adults and two children.
  3. Complete the matrix equation below to show that purchasing a family ticket could give families a saving of $10.  (1 mark)

     \(\displaystyle{\begin {bmatrix} 0 &2&2 \end {bmatrix} \times  N - \begin{bmatrix} \rule{1cm}{0.25mm} & \rule{1cm}{0.25mm} & \rule{1cm}{0.25mm} \end {bmatrix} \times N = \left[ 10\right]}\)

  1. On the opening night, the circus sold 204 family tickets, 162 adult tickets and 176 child tickets.
  2. The owners of the circus want a 3 × 1 product matrix that displays the revenue for each ticket type: family, adult and child.
  3. This product matrix can be achieved by completing the following matrix multiplication.

\(K \times N=\begin{bmatrix}
7344 \\
2430 \\
1408
\end{bmatrix}\)

  1. Write down matrix \(K\).  (1 mark)

Show Answers Only

a.    \(n_{31}\)

b.    \( \begin{bmatrix} 0 & 2 & 2 \end{bmatrix} \times \begin{bmatrix} 36 \\ 15 \\ 8 \end{bmatrix} – \ \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}  \begin{bmatrix} 36 \\ 15 \\ 8 \end{bmatrix} \)

\( = [0 \times 36 + 2 \times 15 + 2 \times 8]-[1 \times 36 + 0 \times 15 + 0 \times 8] \)

\[ = \left[\begin{array}{c} 46 \end{array}\right]-\left[\begin{array}{c} 36 \end{array}\right] \]

\[ = \left[\begin{array}{c} 10 \end{array}\right] \]

c.    \( \begin{bmatrix}204 & 0 & 0 \\ 0 & 162 & 0 \\ 0 & 0 & 176\end{bmatrix} \begin{bmatrix}36 \\ 15 \\ 8\end{bmatrix} = \begin{bmatrix}204 \times 36 \\ 162 \times 15 \\ 176 \times 8\end{bmatrix} = \begin{bmatrix}7344 \\ 2430 \\ 1408\end{bmatrix}\)

Show Worked Solution

a.    \(\text{Cost of one child ticket is in row 3, column 1}\)

\(\Rightarrow n_{31}\)

♦ Mean mark (a) 48%.

 
b.    \( \begin{bmatrix} 0 & 2 & 2 \end{bmatrix} \times \begin{bmatrix} 36 \\ 15 \\ 8 \end{bmatrix} – \ \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}  \begin{bmatrix} 36 \\ 15 \\ 8 \end{bmatrix} \)

\( = [0 \times 36 + 2 \times 15 + 2 \times 8]-[1 \times 36 + 0 \times 15 + 0 \times 8] \)

\[ = \left[\begin{array}{c} 46 \end{array}\right]-\left[\begin{array}{c} 36 \end{array}\right] \]

\[ = \left[\begin{array}{c} 10 \end{array}\right] \]

♦ Mean mark (b) 44%.
c.    \( \begin{bmatrix}204 & 0 & 0 \\ 0 & 162 & 0 \\ 0 & 0 & 176\end{bmatrix} \begin{bmatrix}36 \\ 15 \\ 8\end{bmatrix} = \begin{bmatrix}204 \times 36 \\ 162 \times 15 \\ 176 \times 8\end{bmatrix} = \begin{bmatrix}7344 \\ 2430 \\ 1408\end{bmatrix}\)
♦♦ Mean mark 34%.

Matrices, GEN1 2022 VCAA 8 MC

Two types of computers - laptops `(L)` and desktops `(D)` - can be serviced by Henry `(H)`, Irvine `(I)` or Jean `(J)`.

Matrix `N` shows the time, in minutes, it takes each person to service a laptop and a desktop.

`{:(qquadqquadquad\ LquadqquadD),(N = [(18,8),(10,17),(12,9)]{:(H),(I),(J):}):}`

Matrix `Q` shows the number of laptops and desktops in four different departments: marketing `(M)`, advertising `(A)`, publishing `(P)` and editing `(E)`.

`{:(qquadqquadquad\ LquadqquadD),(Q = [(6,8),(4,7),(5,5),(10,12)]{:(M),(A),(P),(E):}):}`

A calculation that determines the total time that it would take each of Henry, Irvine or Jean, working alone, to service all the laptops and desktops in all four departments is

  1. `[1\ 1\  1\  1]×(Q×N^T)`
  2. `(Q×N^T)×[(1),(1),(1)]`
  3. `(N×Q^T)×Q`
  4. `[(1,0,0),(0,1,0),(0,0,1)]×N×Q^T`
  5. `[1\ 1\ 1\ 1]×Q×N^T×[(1),(1),(1)]`
Show Answers Only

`A`

Show Worked Solution

`text{Required information requires a 3 × 1 or a 1 × 3 matrix.}`

`text{By elimination}`

`text{Option A: [1 × 4] × ([4 × 2][2 × 3]) → [1 × 3] }`

`text{Option B: ([4 × 2][2 × 3]) × [3 × 1] → [4 × 1] (eliminate B)}`

`text{Option C: ([3 × 2][2 × 4]) × [4 × 2] → [3 × 2] (eliminate C)}`

`text{Option D: [3 × 3][3 × 2][2 × 4] → [3 × 4] (eliminate D)}`

`text{Option E: [1 × 4][4 × 2][2 × 3][3 × 1] → [1 × 1] (eliminate E)}`

`=>A`

Matrices, GEN1 2022 VCAA 1-2 MC

A bike rental business rents road bikes \((R)\) and mountain bikes \((M)\) in three sizes: child \((C)\), junior \((J)\) and adult \((A)\).

Matrix \(B\) shows the daily rental cost, in dollars, for each type of bike.

\begin{aligned} \\
& \quad  R \ \quad \ \  M \\
B = & \begin{bmatrix}
80 & 95  \\
110 & 120 \\
120 & 125
\end{bmatrix}\begin{array}{l}
C\\
J\\
A
\end{array}
\end{aligned}

 
The element in row \(i\) and column \(j\) in matrix \(B\) is \(b_{ij}\).

 
Question 1

The daily cost of renting an adult mountain bike is shown in element

  1. \(b_{12}\)
  2. \(b_{21}\)
  3. \(b_{23}\)
  4. \(b_{31}\)
  5. \(b_{32}\)

 
Question 2

On Sundays, the business increases the daily rental price for each type of bike by 10%.

To determine the rental cost for each type of bike on a Sunday, which one of the following matrix calculations needs to be completed?

  1. \(0.01B\)
  2. \(0.1B\)
  3. \(1.01B\)
  4. \(1.1B\)
  5. \(11B\)
Show Answers Only

\(\text{Question 1:}\ E\)

\(\text{Question 2:}\ D\)

Show Worked Solution

\(\text{Question 1}\)

\(\text{Adult Mountain Bike is in Row 3, Column 2}\ = b_{32}\)

\(\Rightarrow E\)
 

\(\text{Question 2}\)

\(\text{Increase of 10% to each type of bike rental charge.}\)

\(\text{Each element increases by 10%.}\)

\(\text{i.e. Multiply each element by a factor of 1.1.}\)

\(\Rightarrow D\)

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 2018 VCAA 2

The Westhorn Council must prepare roads for expected population changes in each of three locations: main town `(M)`, villages `(V)` and rural areas `(R)`.

The population of each of these locations in 2018 is shown in matrix  `P_2018`  below.

`P_2018 = [(2100),(1800),(1700)]{:(M),(V),(R):}`

The expected annual change in population in each location is shown in the table below.
       

  1. Write down matrix  `P_2019`, which shows the expected population in each location in 2019.   (1 mark)

  2. The expected population in each of the three locations in 2019 can be determined from the matrix product.
  3. `qquad qquad P_2019 = F xx P_2018,` where `F` is a diagonal matrix.
  4. Write down matrix  `F`.   (1 mark)

Show Answers Only
  1. `P_2019 = [(1.04 xx 2100),(0.99 xx 1800),(0.98 xx 1700)] = [(2184),(1782),(1666)]`
  2. `F = [(1.04, 0, 0),(0, 0.99, 0),(0, 0, 0.98)]`
Show Worked Solution

a.   `P_2019 = [(1.04 xx 2100),(0.99 xx 1800),(0.98 xx 1700)] = [(2184),(1782),(1666)]`

♦ Mean mark part (b) 40%.
COMMENT: Many students included 0.04, -0.01 and -0.02 in this matrix. Know why this is incorrect!

 
b.   `F = [(1.04, 0, 0),(0, 0.99, 0),(0, 0, 0.98)]`

MATRICES, FUR2 2018 VCAA 1

A toll road is divided into three sections, `E, F` and `G`.

The cost, in dollars, to drive one journey on each section is shown in matrix `C` below.

`C = [(3.58),(2.22),(2.87)]{:(E),(F),(G):}`

  1. What is the cost of one journey on section `G`?   (1 mark)

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

  3. One day Kim travels once on section `E` and twice on section `G`.
  4. His total toll cost for this day can be found by the matrix product  `M xx C`.
  5. Write down the matrix  `M`.   (1 mark)

Show Answers Only
  1. `$2.87`
  2. `text(Order:)\ 3 xx 1`
  3. `M = [(1, 0, 2)]`
Show Worked Solution

a.   `$2.87`
 

b.   `text(Order:)\ 3 xx 1`
 

c.   `M = [(1, 0, 2)]`

MATRICES, FUR2 2017 VCAA 1

A school canteen sells pies (`P`), rolls (`R`) and sandwiches (`S`).

The number of each item sold over three school weeks is shown in matrix `M`.

`{:(qquadqquadqquadquadPqquadRqquadS),(M = [(35,24,60),(28,32,43),(32,30,56)]{:(text(week 1)),(text(week 2)),(text(week 3)):}):}` 

  1. In total, how many sandwiches were sold in these three weeks?   (1 mark)

  2. The element in row `i` and column `j` of matrix `M` is `m_(ij)`.
  3. What does the element `m_12` indicate?  (1 mark)

  4. Consider the matrix equation

    `[(35,24,60),(28,32,43),(32,30,56)] xx [(a),(b),(c)] = [(491.55),(428.00),(487.60)]`

    where `a` = cost of one pie, `b` = cost of one roll and `c` = cost of one sandwich.
  5.  i. What is the cost of one sandwich?   (1 mark)

The matrix equation below shows that the total value of all rolls and sandwiches sold in these three weeks is $915.60

`L xx [(491.55),(428.00),(487.60)] = [915.60]`

Matrix `L` in this equation is of order `1 × 3`.

  1. ii. Write down matrix `L`.   (1 mark)

Show Answers Only
  1. `159`
  2. `text(It represents the number of rolls sold in week 1.)`
    1. `$3.80`
    2. `text(Matrix)\ L = [0,1,1]`
Show Worked Solution
a.    `text(Total sandwiches)` `= 60 + 43 + 56`
    `= 159`

 
b.   `m_12 = 24`

`text(It represents the number of rolls sold in week 1.)`
 

c.i.    `[(a),(b),(c)]` `= [(35,24,60),(28,32,43),(32,30,56)]^(−1)[(491.55),(428.00),(487.60)]`
    `= [(4.65),(4.20),(3.80)]`

 
`:.\ text(C)text(ost of 1 sandwich = $3.80)`
 

c.ii.   `text(Matrix)\ L = [0,1,1]`

MATRICES, FUR2 2016 VCAA 1

A travel company arranges flight (`F`), hotel (`H`), performance (`P`) and tour (`T`) bookings.

Matrix `C` contains the number of each type of booking for a month.

`C = [(85),(38),(24),(43)]{:(F),(H),(P),(T):}`

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

A booking fee, per person, is collected by the travel company for each type of booking.

Matrix `G` contains the booking fees, in dollars, per booking.

`{:((qquadqquadquadF,\ H,\ P,\ T)),(G = [(40,25,15,30)]):}`

  1.  i. Calculate the matrix product  `J = G × C`.   (1 mark)

  2. ii. What does matrix `J` represent?   (1 mark) 

Show Answers Only
  1. `4 xx 1`
    1. `[6000]`
    2. `J\ text(represents the total booking fees for the travel)`
      `text(company in the given month.)`
Show Worked Solution

a.   `text(Order:)\ 4 xx 1`
 

b.i.    `J = [(40,25,15,30)][(85),(38),(24),(43)]= [6000]`

 
b.ii.  `J\ text(represents the total booking fees for the)`

♦ Mean mark 42%.

 `text(travel company in the given month.)`

MATRICES, FUR2 2006 VCAA 1

A manufacturer sells three products, `A`, `B` and `C`, through outlets at two shopping centres, Eastown (`E`) and Noxland (`N`). 

The number of units of each product sold per month through each shop is given by the matrix `Q`, where

`{:((qquadqquadqquad\ A,qquadquadB,qquad\ C)),(Q=[(2500,3400,1890),(1765,4588,2456)]{:(E),(N):}):}`

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

The matrix `P`, shown below, gives the selling price, in dollars, of products `A`, `B`, `C`.

`P = [(14.50),(21.60),(19.20)]{:(A),(B),(C):}`

  1.   i. Evaluate the matrix `M`, where `M = QP`.   (1 mark)

  2.  ii. What information does the elements of matrix `M` provide?   (1 mark)

  3. Explain why the matrix `PQ` is not defined.   (1 mark)

Show Answers Only
  1. `2 xx 3`
    1. `M = QP = [(135\ 320.5),(171\ 848.5)]`
    2. `text(The total of selling products)\ A, B,and C`
      `text(at each of Eastown and Noxland.)`
  3. `PQ\ text(is not defined because the number of)`
    `text(columns in)\ P !=\ text(the number of rows in)\ Q.`
Show Worked Solution

a.   `2 xx 3`
 

b.i.    `M` `= QP`
    `= [(2500,3400,1890),(1765,4588,2456)][(14.50),(21.60),(19.20)]`
    `= [(135\ 320.5),(171\ 848.5)]`

 
b.ii.  `text(The total revenue from selling products)\ A, B,`

   `text(and)\ C\ text(at each of Eastown and Noxland.)`
 

c.   `PQ\ text(is not defined because the number of)`

`text(columns in)\ P !=\ text(the number of rows in)\ Q.`

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 2009 VCAA 1

Three types of cheese, Cheddar (`C`), Gouda (`G`) and Blue (`B`), will be bought for a school function.

The cost matrix `P` lists the prices of these cheeses, in dollars, at two stores, Foodway and Safeworth.
 

`P = [(6.80, 5.30, 6.20),(7.30, 4.90, 6.15)]{:(text(Foodway)),(text(Safeworth)):}`
 

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

The number of packets of each type of cheese needed is listed in the quantity matrix `Q`.
 

`Q = [(8),(11),(3)]{:(C),(G),(B):}`
 

    1. Evaluate the matrix  `W = PQ`.   (1 mark)
    2. At which store will the total cost of the cheese be lower?   (1 mark)

Show Answers Only

  1. `2 xx 3`
    1. `W = PQ = [(131.30),(130.75)]`
    2. `text(Safeworth)`

Show Worked Solution

a.   `2 xx 3`

 

b.i.    `W` `=PQ`
    `= [(6.80,5.30,6.20),(7.30,4.90,6.15)][(8),(11),(3)]`
    `= [(131.30),(130.75)]`

 

b.ii.   `text(Safeworth)`

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 2

To reduce the number of insects in a wetland, the wetland is sprayed with an insecticide.

The number of insects (`I`), birds (`B`), lizards (`L`) and frogs (`F`) in the wetland that has been sprayed with insecticide are displayed in the matrix `N` below.
 

`{:((qquadqquadqquadqquadI,qquadquad B,qquadL,\ qquadF)),(N = [(100\ 000, 400,1000,800)]):}`
 

Unfortunately, the insecticide, that is used to kill the insects can also kill birds, lizards and frogs. The proportion of insects, birds, lizards and frogs that have been killed by the insecticide are displayed in the matrix `D` below.
 

`{:(qquadqquadqquadquadquadtext(alive before spraying)),((qquadqquadqquadqquadI,qquad\ B,qquad\ L,qquad\ F)),(D = [(0.995,0,0,0),(0,0.05,0,0),(0,0,0.025,0),(0,0,0,0.30)]{:(I),(B),(L),(F):}{:qquadtext(dead after spraying):}):}`
 

  1. Evaluate the matrix product  `K = ND`.   (1 mark)

  2. Use the information in matrix `K` to determine the number of birds that have been killed by the insecticide.   (1 mark)

  3. Evaluate the matrix product  `M = KF`, where `F = [(0),(1),(1),(1)]`.   (1 mark)

  4. In the context of the problem, what information does matrix `M` contain?   (1 mark)

Show Answers Only
  1. `K = [(99\ 500,20,25,240)]`
  2. `20`
  3. `M = [285]`
  4. `M\ text(contains the combined number)`
    `text(of birds, lizards and frogs that died.)`
Show Worked Solution
a.    `K` `= ND`
    `= [(99\ 500,20,25,240)]`
MARKER’S COMMENT: In part (a), separating matrix elements by commas or dots is not correct notation.

 
b.   `text(Birds dead after spraying) = 20`
 

c.    `M` `= KF`
    `= [(99\ 500,20,25,240)][(0),(1),(1),(1)]`
    `= [0 + 20 + 25 + 240]`
    `= [285]`
     
MARKER’S COMMENT: The ability of many students to interpret the result of a matrix product was poor.

d.   `text(Matrix)\ M\ text(contains the combined number)`

`text(of birds, lizards and frogs that died.)`

 

MATRICES, FUR1 2008 VCAA 3 MC

The cost prices of three different electrical items in a store are $230, $290 and $310 respectively.

The selling price of each of these three electrical items is 1.3 times the cost price plus a commission of $20 for the salesman.

A matrix that lists the selling price of each of these three electrical items is determined by evaluating

A.  `1.3 xx [(230), (290), (310)] + [20]`

B.  `1.3 xx [(230), (290), (310)] + 1.3 xx 20`

C.  `1.3 xx [(230), (290), (310)] + [(20), (20), (20)]`

D.  `1.3 xx [(230), (290), (310)] + 1.3 xx [(20), (20), (20)]`

E.  `1.3 xx [(230 + 20), (290 + 20), (310 + 20)]`

Show Answers Only

`C`

Show Worked Solution

`=>   C`

MATRICES, FUR1 2010 VCAA 6 MC

Vince, Nev and Rani all service office equipment.

The matrix `T` shows the time that it takes (in minutes) for each of Vince (V), Nev (N) and Rani (R) to service a photocopier (P) a fax machine (F) and a scanner (S).
 

`{:(qquad qquad qquad {:(V,\ N,\ R):}), (T = [(12, 15, 14),(8, 7, 8), (20, 19, 17)] {:(P), (F), (S):}):}`
 

The matrix `U` below displays the number of photocopiers, fax machines and scanners to be serviced in three schools, Alton (A), Borton (B) and Carlon (C).
 

`{:(qquad qquad qquad {:(P, F, S):}), (U = [(5,\ 3,\ 2),(4,\ 4,\ 3), (6,\ 1,\ 2)] {:(A), (B), (C):}):}`
 

A matrix that displays the time that it would take each of Vince, Nev and Rani, working alone, to service the photocopiers, fax machines and scanners in each of the three schools is

A.  `[(17, 18, 16), (12, 11, 11), (26, 20, 19)]` B.  `[(204, 110, 97), (116, 60, 53), (278, 153, 131)]`
       
C.  `[(124, 134, 128), (140, 145, 139), (120, 135, 126)]` D.  `[(7, 12, 12), (4, 3, 5), (14, 18, 15)]`
       
E.  `[(60, 15, 28), (32, 35, 24), (120, 19, 34)]`    

 

Show Answers Only

`C`

Show Worked Solution

`text(Matrix)\ \ UT\ \ text(gives the information.)`

♦♦♦ Mean mark 19%.
MARKER’S COMMENT: Most students incorrectly calculated matrix `TU`. Make sure you know why this is incorrect!
`UT` `= [(5, 3, 2), (4, 4, 3), (6, 1, 2)] [(12, 15, 14), (8, 7, 8), (20, 19, 17)]`
  `= [(124, 134, 128), (140, 145, 139), (120, 135, 126)]`

 
`=>   C`

MATRICES, FUR1 2010 VCAA 2 MC

Peter bought only apples and bananas from his local fruit shop.

The matrix

`{:(qquad qquad quad {:\ text(A  B):}), (N = [(3, 4)]):}`
 

lists the number of apples (A) and bananas (B) that Peter bought

The matrix
 

`C = [(0.37), (0.43)] {:(A), (B):}`
 

lists the cost (in dollars) of one apple and one banana respectively

The matrix product, `NC`, gives

  1. the total amount spent by Peter on the fruit that he bought.
  2. the total number of pieces of fruit that Peter bought.
  3. the individual amounts that Peter spent on apples and bananas respectively.
  4. the total number of pieces of fruit that Peter bought and the total amount that he spent.
  5. the individual number of apples and bananas that Peter bought and the individual amounts that Peter spent on these apples and bananas respectively.
Show Answers Only

`A`

Show Worked Solution

`NC\ \ text(is a 1 × 1 matrix that gives the total amount)`

`text(spent on both fruits.)`

`=>   A`

MATRICES, FUR1 2012 VCAA 6 MC

 The table below shows the number of classes and the number of students in each class at each year level in a secondary school.
 

MATRICES, FUR1 2012 VCAA 6 MC1
  

Let   `F= [1 quad 1 quad 1 quad 1], \ G= [(1),(1),(1),(1)],\ M= [7 quad 5 quad 6 quad 4],\ N= [(7),(5),(6),(4)],\ P= [(22,0,0,0), (0,20,0,0), (0,0,18,0), (0,0,0,24)]`
 

A matrix product that displays the total number of students in Years 9 – 12 at this school is

A.   `M xx P xx F`

B.   `P xx G xx M`

C.   `F xx P xx N`

D.   `P xx N xx F`

E.   `F xx N xx P`

Show Answers Only

`C`

Show Worked Solution

`text(Total student matrix will produce)`

`text(a 1 × 1 matrix.)`

`text(Consider)\ C,`

   `F` `xx`  `P` `xx`   `N` `=`      `FPN`
`1 xx 4`    `4 xx 4`     `4 xx 1`        `1 xx 1`

 

`A, B\ text(and)\ E\ text(produce undefined matrices)`

`text(and)\ D\ text(produces a 4 × 4.)`

`rArr C`

MATRICES, FUR1 2013 VCAA 7 MC

A school has three computer classes, A, B and C. There are 15 students in each class.

Each student is given a mark out of 100 based on their performance in a test.

Matrix `M` below displays the marks obtained by these 45 students, listed by class.
 

`M = [(56,78,79,43,67,56,80,85,75,89,55,64,95,34,63), (90,45,56,65,76,79,27,45,69,73,70,63,65,34,59), (76,76,89,47,50,66,68,89,88,90,45,67,78,45,87)]{:(A),(B),(C):} quad text (class)`
 

 Two other matrices, `S` and `R`, are defined below.
 

`S = [(1),(1),(1),(1),(1),(1),(1),(1),(1),(1),(1),(1),(1),(1),(1)] quad text(and) quad R = [(1,1,1)]`
 

Which one of the following matrix expressions can be used to generate a matrix that displays the mean mark obtained for each class?

A.   `1/45M`

B.   `1/3R×M`

C.   `1/3R×M×S`

D.   `1/15M×S`

E.   `1/15S×R×M`

Show Answers Only

`D`

Show Worked Solution
   `M`  `xx`   `S` `=`     `MS`
`3 xx 15`     `15 xx 1`       `3 xx 1`

 
`MS\ text(represents the total scores in each class.)`
 

`:. 1/15 MS\ text(is the average score in each class.)`

`rArr D`

MATRICES, FUR1 2006 VCAA 5 MC

A company makes Regular (`R`), Queen (`Q`) and King (`K`) size beds. Each bed comes in either the Classic style or the more expensive Deluxe style.

The price of each style of bed, in dollars, is listed in a price matrix `P`, where
 

`{:(qquadqquadqquad\ RquadqquadQquadqquadK),(P = [(145, 210, 350), (185, 270, 410)]{: (text (Classic)), (text (Deluxe)) :}):}`
 

The company wants to increase the price of all beds.

A new price matrix, listing the increased prices of the beds, can be generated from `P` by forming a matrix product with the matrix, `M`, where
 

`M = [(1.2,0), (0, 1.35)]`
 

This new price matrix is

MATRICES, FUR1 2006 VCAA 5 MC ab 1

MATRICES, FUR1 2006 VCAA 5 MC cd 1

MATRICES, FUR1 2006 VCAA 5 MC e

Show Answers Only

`D`

Show Worked Solution
`MP` `= [(1.2,0),(0,1.35)][(145,210,350),(185,270,410)]`
  `= [(174,252,420),(249.75,364.50,553.50)]`

 
`rArr D`