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Matrices, GEN2 2025 VCAA 14

An early learning centre runs seven different activities during its 40-day holiday program.

The activities are cooking \((C)\), drama \((D)\), gardening \((G)\), lunch \((L)\), music \((M)\), reading \((R)\) and sport \((S)\).

The timetabled order of the activities for day one of the holiday program is shown below.

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \text{9 am} \ \ \rule[-1ex]{0pt}{0pt} & \text{10 am} \rule[-1ex]{0pt}{0pt} & \text{11 am} \rule[-1ex]{0pt}{0pt} & \text{12 pm} \rule[-1ex]{0pt}{0pt} & \ \ \text{1 pm} \ \ \rule[-1ex]{0pt}{0pt} & \ \ \text{2 pm} \ \ \rule[-1ex]{0pt}{0pt} & \ \ \text{3 pm} \ \ \\
\hline
\rule{0pt}{2.5ex} \textit{C} \rule[-1ex]{0pt}{0pt} & \textit{D} \rule[-1ex]{0pt}{0pt} & \textit{G} \rule[-1ex]{0pt}{0pt} & \textit{L} \rule[-1ex]{0pt}{0pt} & \textit{M} \rule[-1ex]{0pt}{0pt} & \textit{R} \rule[-1ex]{0pt}{0pt} & \textit{S}\\
\hline
\end{array}

The timetabled order of the activities for day one is also shown in matrix \(X\) below.

\begin{aligned} 
X = & \begin{bmatrix}
C  \\
D \\
G \\
L \\
M \\
R\\
S \\
\end{bmatrix}
\end{aligned}

Matrix \(P\), shown below, is a permutation matrix used to determine the timetabled order of activities from one day to the next.

\begin{aligned} 
P = & \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\end{aligned}

A column matrix containing the timetabled order of activities on one day is multiplied by matrix \(P\) to determine the timetabled order of activities for the next day.

  1. State the activities that are always held at the same time on each day of the program.  (1 mark)
  2. Determine the timetabled order of the seven activities on day three of the program.  (1 mark)
  3. \(P^4\) is an identity matrix.
  4. Explain what this means for the timetabled order of the activities over the 40-day holiday program.  (1 mark)
Show Answers Only

a.    \(\text{Gardening, lunch, music.}\)

b.    \(\text{Drama, cooking, gardening, lunch, music, sport, reading.}\)

c.    \(\text{Order of activities rotate on a 4 day cycle.}\)

\(\text{Over 40 days, there will be 10 cycles of activities}\)

Show Worked Solution

a.    \(\text{Activities held af the same time:}\)

\(\Rightarrow \ \text{correspond to 1’s in leading diagonal}\)

\(\text{Gardening, lunch, music.}\)
 

b. 

\begin{aligned}X_2=P \times X=\begin{bmatrix}
R \\
S \\
G \\
L \\
M \\
D \\
C
\end{bmatrix}, \quad X_3=P \times X_2=\begin{bmatrix}
D \\
C \\
G \\
L \\
M \\
S \\
R
\end{bmatrix}
\end{aligned}

\(\text{Order on day 3:}\)

\(\text{Drama, cooking, gardening, lunch, music, sport, reading.}\)

♦♦ Mean mark (b) 27%.

c.    \(P^4 \ \Rightarrow \ \text{identity matrix}\)

\(\text{Order of activities rotate on a 4 day cycle.}\)

\(\text{Over 40 days, there will be 10 cycles of activities}\)

♦♦♦ Mean mark (c) 11%.

Matrices, GEN1 2024 NHT 30 MC

Matrix \(R\) is a column matrix.

\begin{align*}
R=\begin{bmatrix}
T \\
A \\
L \\
L \\
Y
\end{bmatrix}
\end{align*}

A permutation matrix, \(P\), is multiplied by matrix \(R\) to form the product matrix  \(Q=P R\).

If \(Q\) is equal to \(R\), how many different permutation matrices could have been used?

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
Show Answers Only

\(B\)

Show Worked Solution

\(Q=P R=R\)

\(\text{The identity matrix satisfies the equation (see \(P_1\) below).}\)

\(\text{Matrix \(R\) also remains the same if \(e_{31}\) and \(e _{41}\) are swapped (see \(P_2\) below).}\)
 

\(P_1=I=\begin{bmatrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0\end{bmatrix} \quad \text{or} \quad P_2=\begin{bmatrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\)

\(\Rightarrow B\)

MATRICES, FUR1 2018 VCAA 4 MC

Matrix `P` is a 4 × 4 permutation matrix.

Matrix `W` is another matrix such that the matrix product `P × W` is defined.

This matrix product results in the entire first and third rows of matrix `W` being swapped.

The permutation matrix `P` is

A.  `[(0,0,0,1),(0,1,0,0),(0,0,1,0),(0,0,0,1)]` B.  `[(0,0,1,0),(0,1,0,0),(1,0,0,0),(0,0,0,1)]` C.  `[(1,0,0,0),(0,0,0,0),(1,0,0,0),(0,0,0,0)]`
           
D.  `[(1,0,0,0),(0,0,0,0),(0,0,1,0),(0,0,0,0)]` E.  `[(1,0,0,0),(0,1,0,0),(1,0,0,0),(0,0,0,1)]`    
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`B`

Show Worked Solution

`text(By trial and error:)`

`text(S)text(ince)\ P xx W\ \ text(is defined,)`

`=>\ W\ text(can be a 4 × 1 matrix)`
 

`text(Consider option)\ B,`

`[(0,0,1,0),(0,1,0,0),(1,0,0,0),(0,0,0,1)][(a),(b),(c),(d)] = [(c),(b),(a),(d)]`
 

`=> B`

MATRICES, FUR1 2017 VCAA 4 MC

A permutation matrix, `P`, can be used to change `[(F),(E),(A),(R),(S)]` into `[(S),(A),(F),(E),(R)]`.

Matrix `P` is

A.

`[(0,0,1,0,1),(0,0,1,1,0),(1,1,0,0,0),(0,1,0,0,1),(1,0,0,1,0)]`

 

 B.

`[(0,0,0,1,0),(0,0,1,0,0),(0,1,0,0,0),(0,0,0,0,1),(1,0,0,0,0)]`

 
C.

`[(0,0,0,0,1),(0,0,1,0,0),(1,0,0,0,0),(0,1,0,0,0),(0,0,0,1,0)]`

 

 D.

`[(1,0,0,0,1),(0,1,1,0,0),(1,0,1,0,0),(0,1,0,1,0),(0,0,0,1,1)]`

 
E.

`[(0,0,0,0,1),(0,0,1,0,0),(0,1,0,0,0),(1,0,0,0,0),(0,0,0,1,0)]`

 

    
Show Answers Only

`C`

Show Worked Solution

`[(0,0,0,0,1),(0,0,1,0,0),(1,0,0,0,0),(0,1,0,0,0),(0,0,0,1,0)][(F),(E),(A),(R),(S)] = [(S),(A),(F),(E),(R)]`

`=> C`