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Matrices, GEN2 2023 VCAA 10

Within the circus, there are different types of employees: directors \((D)\), managers \((M)\), performers \((P)\) and sales staff \((S).\) Customers \((C)\) attend the circus.

Communication between the five groups depends on whether they are customers or employees, and on what type of employee they are.

Matrix \(G\) below shows the communication links between the five groups.

\begin{aligned}
&\quad \quad \quad\quad \quad \quad\quad \quad \quad \ \ \textit{receiver}\\
&\quad \quad\quad \quad \quad\quad \quad \quad D \ \ M \ \  P \ \ \  S \ \ \  C \\
& G=\textit{sender} \quad \begin{array}{ccccc}
D\\
M\\
P\\
S\\
C
\end{array}
\begin {bmatrix}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0
\end{bmatrix}\\
&
\end{aligned}

In this matrix:

    • The ' 1 ' in row \(D\), column \(M\) indicates that the directors can communicate directly with the managers.
    • The ' 0 ' in row \(P\), column \(D\) indicates that the performers cannot communicate directly with the directors.
  1. A customer wants to make a complaint to a director.
  2. What is the shortest communication sequence that will successfully get this complaint to a director?  (1 mark)

  3. Matrix \(H\) below shows the number of two-step communication links between each group. Sixteen elements in this matrix are missing.

\begin{aligned}
&\quad \quad \quad\quad \quad \quad\quad \quad \quad \ \ \textit{receiver}\\
&\quad \quad\quad \quad \quad\quad \quad \quad D \quad M \quad P \quad \ S \quad \ C \\
& H=\textit{sender} \quad \begin{array}{ccccc}
D\\
M\\
P\\
S\\
C
\end{array}
\begin {bmatrix} {\displaystyle}
1 & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} \\
0 & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} \\
1 & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} \\
1 & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} & \rule{0.5cm}{0.15mm} \\
0 & 1 & 0 & 0 & 1
\end{bmatrix}\\
&
\end{aligned}

  2.  i. Complete matrix \(H\) above by filling in the missing elements.  (1 mark)
  4. ii. What information do elements \(g_{21}\) and \(h_{21}\) provide about the communication between the circus employees?  (1 mark)

Show Answers Only

a.    \(C\ S\ M\ D\)
 

b.i.  \( \quad \) \(\begin{aligned} & \begin{array}{cccccc}\quad \ \ \ D \ \ M \ \ \ P  \ \ \ S \ \ \ C\end{array} \\ &
\begin{array}{c}D \\ M \\P \\ S \\ C\end{array}
\begin{bmatrix}
1 & 2 & 1 & 2 & 2 \\
0 & 3 & 1 & 2 & 2 \\
1 & 0 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 & 1
\end{bmatrix}\\ & \end{aligned}\)

b.ii. \(g_{21}\ \text{indicates managers can communicate directly with directors.}\)

\(h_{21}\ \text{indicates managers, however, cannot communicate with}\)

\(\text{directors through another person who speaks directly to a director.}\)

Show Worked Solution

a.    \(C\ S\ M\ D\)
 

b.i.  \( \quad \) \(\begin{aligned} & \begin{array}{cccccc}\quad \ \ \ D \ \ M \ \ \ P  \ \ \ S \ \ \ C\end{array} \\ &
\begin{array}{c}D \\ M \\P \\ S \\ C\end{array}
\begin{bmatrix}
1 & 2 & 1 & 2 & 2 \\
0 & 3 & 1 & 2 & 2 \\
1 & 0 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 & 1
\end{bmatrix}\\ & \end{aligned}\)

b.ii. \(g_{21}\ \text{indicates managers can communicate directly with directors.}\)

\(h_{21}\ \text{indicates managers, however, cannot communicate with}\)

\(\text{directors through another person who speaks directly to a director.}\)

♦♦♦ Mean mark (b)(ii) 25%.

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 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 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 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 5 MC

`A`, `B`, `C`, `D` and `E` are five intersections joined by roads as shown in the diagram below.

Some of these roads are one-way only.
 

MATRICES, FUR1 2009 VCAA 5 MC

The matrix below indicates the direction that cars can travel along each of these roads.

In this matrix

    • 1 in column `A` and row `B` indicates that cars can travel directly from `A` to `B`
    • 0 in column `B` and row `A` indicates that cars cannot travel directly from `B` to `A` (either it is a one-way road or no road exists).
       

`{:(text(from intersection)),({:quad\ Aquad\ Bquad\ Cquad\ Dquad\ E:}),([(0,0,0,0,0),(1,0,0,0,0),(0,1,0,1,1),(1,0,0,0,0),(0,0,1,1,0)]):}{:(),(),(A),(B),(Cqquadtext(to intersection)),(D),(E):}`
 

Cars can travel in both directions between intersections

A.   `A` and `D`

B.   `B` and `C`

C.   `C` and `D`

D.   `D` and `E`

E.   `C` and `E`

Show Answers Only

`E`

Show Worked Solution

`e_(CE) = e_(EC) = 1`

`:.\ text(Cars can travel both ways)`

`text(between)\ E\ text(and)\ C.` 

`=>  E`