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Matrices, GEN1 2024 NHT 32 MC

An online shop offers monthly subscriptions for protein powder.

The shop offers protein powder in three flavours: vanilla \((V)\), chocolate \((C)\) and malt \((M)\).

Let \(P_n\) be the state matrix that shows the expected number of subscribers for each flavour \(n\) months after sales of the protein powder began.

The expected number of subscribers for each flavour can be determined by the matrix recurrence rule

\(P_{n+1}=T P_n+K\)

where

\begin{aligned}
& \quad \quad \quad \ \text { this month }\\
& \quad \quad \quad \ V \quad \ \ C \quad \ M \\ 
& T=\begin{bmatrix}0.2 & 0.2 & 0.1 \\
0.4 & 0.2 & 0.1 \\
0.4 & 0.6 & 0.8
\end{bmatrix} \begin{array}{l}
V \\ C\\ M
\end{array}
\ \text{next month} \quad \text { and } \quad K=\begin{bmatrix} 93 \\ 59 \\ 9\end{bmatrix}\begin{array}{l}V \\ C \\ M \end{array}\end{aligned}

The state matrix, \(P_2\), below shows the expected number of subscribers for each flavour two months after sales began.

\begin{align*}
P_2=\begin{bmatrix}
147 \\
137 \\
199
\end{bmatrix}
\end{align*}

The increase in the expected number of subscribers for vanilla \((V)\) between the initial sales, \(P_0\), and the first month after sales began, \(P_1\), is equal to

  1. 27
  2. 54
  3. 60
  4. 87
  5. 93
Show Answers Only

\(C\)

Show Worked Solution

\(P_{n+1} = TP_{n}+K\ \Rightarrow\ P_2=TP_1+K \ \Rightarrow\ P_1=T^{-1}(P_2-K) \) 
 

\(P_1=\begin{bmatrix}0.2 & 0.2 & 0.1 \\ 0.4 & 0.2 & 0.1 \\ 0.4 & 0.6 & 0.8\end{bmatrix}^{-1}\left(\begin{bmatrix}147 \\ 137 \\ 199\end{bmatrix}-\begin{bmatrix}93 \\ 59 \\ 9\end{bmatrix}\right)=\begin{bmatrix} 120 \\ 98 \\ 104\end{bmatrix}\)
 

\(P_0=\begin{bmatrix}0.2 & 0.2 & 0.1 \\ 0.4 & 0.2 & 0.1 \\ 0.4 & 0.6 & 0.8\end{bmatrix}^{-1}\left(\begin{bmatrix}120 \\ 98 \\ 104\end{bmatrix}-\begin{bmatrix}93 \\ 59 \\ 9\end{bmatrix}\right)=\begin{bmatrix}60 \\ 49 \\ 52\end{bmatrix}\)
 

\(\therefore\ \text{Increase in vanilla subscribers}\ =120-60=60\)

\(\Rightarrow C\)

Matrices, GEN1 2024 VCAA 32 MC

A large sporting event is held over a period of four consecutive days: Thursday, Friday, Saturday and Sunday.

People can watch the event at four different sites throughout the city: Botanical Gardens \((G)\), City Square \((C)\), Riverbank \((R)\) or Main Beach \((M)\).

Let \(S_n\) be the state matrix that shows the number of people at each location \(n\) days after Thursday.

The expected number of people at each location can be determined by the matrix recurrence rule

\(S_{n+1}=TS_n+A\)

\begin{aligned}
& \quad \quad \quad \quad \quad \quad \quad \quad \quad \textit{this day} \\
& \quad \quad \quad \quad \quad \quad \quad \ G \quad \ \  C \quad \ \ R \quad \ \  M \\
& \text{where} \quad T=\begin{bmatrix}
0.4 & 0.2 & 0.4 & 0 \\
0.4 & 0.1 & 0.3 & 0.3 \\
0.1 & 0.4 & 0.1 & 0.2 \\
0.1 & 0.3 & 0.2 & 0.5
\end{bmatrix}\begin{array}{l}
G \\
C \\
R \\
M
\end{array} \text { next day } \quad \text{and}& A=\begin{bmatrix}
300 \\
200\\
100 \\
300
\end{bmatrix}\begin{array}{l}
G \\
C \\
R \\
M
\end{array}
\end{aligned}

 

\begin{aligned} \text{Given the state matrix}& \quad \quad S_3=\begin{bmatrix}
5620\\
6386\\
4892\\
6902
\end{bmatrix}\begin{array}{l}
G \\
C \\
R \\
M
\end{array}
\end{aligned}

the number of people watching the event at the Botanical Gardens \((G)\) from Thursday to Sunday has

  1. decreased by 162
  2. decreased by 212
  3. increased by 124
  4. increased by 696
Show Answers Only

\(D\)

Show Worked Solution
\(S_{n+1}\) \(=TS_n+A\)
\(TS_n\) \(=S_{n+1}-A\)
\(S_n\)  \(=T^{-1}\times \left(S_{n+1}-A\right)\)
♦♦♦ Mean mark 33%.

  
\(\text{Using CAS:}\)
  
\(T =\begin{bmatrix}
\ 0.4 & 0.2 & 0.4 & 0 \\
\ 0.4 & 0.1 & 0.3 & 0.3 \\
\ 0.1 & 0.4 & 0.1 & 0.2 \\
\ 0.1 & 0.3 & 0.2 & 0.5 
\end{bmatrix}\ \rightarrow \ \ T^{-1}=
 \begin{bmatrix}
\ -2.6 & 5.4 & 3.4 & -4.6 \\
\ 0.2 & -0.8 & 3.2 & -0.8 \\
\ 5 & -5 & -5 & 5 \\
\ -1.6 & 1.4 & -0.6 & 1.4   
\end{bmatrix}\)

 

\(\text{Sunday} =S_3=\begin{bmatrix}
\ 5620 \\
\ 6386 \\
\ 4892  \\
\ 6902
\end{bmatrix} \)

\(\text{Saturday} =S_2=T^{-1}\times \left(S_{3}-A\right)=\begin{bmatrix}
\ 5496 \\
\ 6168 \\
\ 4720  \\
\ 6516
\end{bmatrix} \)

\(\text{Friday} =S_1=T^{-1}\times \left(S_{2}-A\right)=\begin{bmatrix}
\ 5832 \\
\ 6076 \\
\ 4120  \\
\ 5972
\end{bmatrix} \)

\(\text{Thursday} =S_0=T^{-1}\times \left(S_{1}-A\right)=\begin{bmatrix}
\ 4924 \\
\ 4732 \\
\ 6540  \\
\ 4904
\end{bmatrix} \)

  
\(5620-4924=696\)

\(\therefore\ \text{Botanical Gardens attendance has increased by 696 people.}\)

\(\Rightarrow D\)

MATRICES, FUR2 2020 VCAA 4

A second market research project also suggested that if the Westmall shopping centre were sold, each of the three centres (Westmall, Grandmall and Eastmall) would continue to have regular shoppers but would attract and lose shoppers on a weekly basis.

Let `R_n` be the state matrix that shows the expected number of shoppers at each of the three centres `n` weeks after Westmall is sold.

A matrix recurrence relation that generates values of `R_n` is

`R_(n+1) = TR_n + B`

`{:(quad qquad qquad qquad qquad qquad qquad qquad text(this week)),(qquad qquad qquad qquad qquad qquad quad \ W qquad quad G qquad quad \ E),(text(where)\ T = [(quad 0.78, 0.13, 0.10),(quad 0.12, 0.82, 0.10),(quad 0.10, 0.05, 0.80)]{:(W),(G),(E):}\ text(next week,) qquad qquad  B = [(-400), (700), (500)]{:(W),(G),(E):}):}`
 

The matrix `R_2` is the state matrix that shows the expected number of shoppers at each of the three centres in the second week after Westmall is sold

`R_2 = [(239\ 060), (250\ 840), (192\ 900)]{:(W),(G),(E):}`

  1. Determine the expected number of shoppers at Westmall in the third week after it is sold.   (1 mark)

  2. Determine the expected number of shoppers at Westmall in the first week after it is sold.   (1 mark)

Show Answers Only
  1. `237\ 966`
  2. `241\ 000`
Show Worked Solution

♦ Mean mark part (a) 50%.
a.   `R_3` `= TR_2 + B`
    `= [(0.78, 0.13, 0.1),(0.12, 0.82, 0.1),(0.10, 0.05, 0.8)][(239\ 060),(250\ 840),(192\ 900)]+[(-400),(700),(500)] = [(237\ 966),(254\ 366),(191\ 268)]`

 
`:. text(Expected Westmall shoppers) = 237\ 966`
 

♦♦♦ Mean mark part (b) 20%.
b.   `R_2` `= TR_1 + B`
  `R_1` `= T^(-1)[R_2-B]`
    `= [(241\ 000), (246\ 000), (195\ 000)]`

 
`:. text(Expected Westmall shoppers) = 241\ 000`