VCE General Maths Matrices 2020 Exam 1 Mini Test

The matrix \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}\) is an example of

• A. a binary matrix.
• B. an identity matrix.
• C. a triangular matrix.
• D. a symmetric matrix.
• E. a permutation matrix.

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Correct Answer: A
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Matrix \(A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \\ 1 & 0 \\ 4 & 5 \end{bmatrix}\) and matrix \(B = \begin{bmatrix} 2 & 0 & 3 & 1 \\ 4 & 5 & 2 & 0 \end{bmatrix}\)

Matrix \(Q = A \times B\).

The element in row \(i\) and column \(j\) of matrix \(Q\) is \(q_{ij}\).

Element \(q_{41}\) is determined by the calculation

• A. \(0 \times 0 + 3 \times 5\)
• B. \(1 \times 1 + 2 \times 0\)
• C. \(1 \times 2 + 2 \times 4\)
• D. \(4 \times 1 + 5 \times 0\)
• E. \(4 \times 2 + 5 \times 4\)

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Correct Answer: E
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Matrices \(P\) and \(W\) are defined below.

\[ P = \begin{bmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{bmatrix} \quad W = \begin{bmatrix} A \\ S \\ T \\ O \\ R \end{bmatrix} \]

If \(P^n \times W = \begin{bmatrix} A \\ S \\ T \\ O \\ R \end{bmatrix}\), the value of \(n\) could be

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

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Correct Answer: D
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In a particular supermarket, the three top-selling magazines are Angel (\(A\)), Bella (\(B\)) and Crystal (\(C\)). The transition diagram below shows the way shoppers at this supermarket change their magazine choice from week to week.

A transition matrix that provides the same information as the transition diagram is

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Correct Answer: C
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The diagram below shows the direct communication links that exist between Sam (\(S\)), Tai (\(T\)), Umi (\(U\)) and Vera (\(V\)). For example, the arrow from Umi to Vera indicates that Umi can communicate directly with Vera.

A communication matrix can be used to convey the same information.

In this matrix:

• a '1' indicates that a direct communication link exists between a sender and a receiver
• a '0' indicates that a direct communication link does not exist between a sender and a receiver.

The communication matrix could be

• A. \( \begin{array}{c@{\hskip 1em}c} \begin{array}{c} \\ \textit{sender} \\ \\ \end{array} & \begin{array}{c} \textit{receiver} \\ \begin{array}{cccc} S & T & U & V \end{array} \\ \left[ \begin{array}{cccc} 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{array} \right] \end{array} \end{array} \begin{array}{c} \\ S \\ T \\ U \\ V \end{array} \)
• B. \( \begin{array}{c@{\hskip 1em}c} \begin{array}{c} \\ \textit{sender} \\ \\ \end{array} & \begin{array}{c} \textit{receiver} \\ \begin{array}{cccc} S & T & U & V \end{array} \\ \left[ \begin{array}{cccc} 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{array} \right] \end{array} \end{array} \begin{array}{c} \\ S \\ T \\ U \\ V \end{array} \)
• C. \( \begin{array}{c@{\hskip 1em}c} \begin{array}{c} \\ \textit{sender} \\ \\ \end{array} & \begin{array}{c} \textit{receiver} \\ \begin{array}{cccc} S & T & U & V \end{array} \\ \left[ \begin{array}{cccc} 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \end{array} \right] \end{array} \end{array} \begin{array}{c} \\ S \\ T \\ U \\ V \end{array} \)
• D. \( \begin{array}{c@{\hskip 1em}c} \begin{array}{c} \\ \textit{sender} \\ \\ \end{array} & \begin{array}{c} \textit{receiver} \\ \begin{array}{cccc} S & T & U & V \end{array} \\ \left[ \begin{array}{cccc} 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{array} \right] \end{array} \end{array} \begin{array}{c} \\ S \\ T \\ U \\ V \end{array} \)
• E. \( \begin{array}{c@{\hskip 1em}c} \begin{array}{c} \\ \textit{sender} \\ \\ \end{array} & \begin{array}{c} \textit{receiver} \\ \begin{array}{cccc} S & T & U & V \end{array} \\ \left[ \begin{array}{cccc} 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 2 & 2 & 2 & 0 \end{array} \right] \end{array} \end{array} \begin{array}{c} \\ S \\ T \\ U \\ V \end{array} \)

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Correct Answer: D
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The element in row \(i\) and column \(j\) of matrix \(M\) is \(m_{ij}\).

\(M\) is a \(3 \times 3\) matrix. It is constructed using the rule \(m_{ij} = 3i + 2j\).

\(M\) is

• A. \(\begin{bmatrix} 5 & 7 & 9 \\ 7 & 9 & 11 \\ 11 & 13 & 15 \end{bmatrix}\)
• B. \(\begin{bmatrix} 5 & 7 & 9 \\ 8 & 10 & 12 \\ 11 & 13 & 15 \end{bmatrix}\)
• C. \(\begin{bmatrix} 5 & 7 & 10 \\ 8 & 10 & 13 \\ 11 & 13 & 16 \end{bmatrix}\)
• D. \(\begin{bmatrix} 5 & 8 & 11 \\ 7 & 10 & 13 \\ 9 & 12 & 15 \end{bmatrix}\)
• E. \(\begin{bmatrix} 5 & 8 & 11 \\ 8 & 11 & 14 \\ 11 & 14 & 17 \end{bmatrix}\)

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Correct Answer: B
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A small shopping centre has two coffee shops: Fatima's (\(F\)) and Giorgio's (\(G\)). The percentage of coffee-buyers at each shop changes from day to day, as shown in the transition matrix \(T\).

\[ T = \begin{array}{c} \begin{array}{cc} F & G \end{array} \\ \begin{bmatrix} 0.85 & 0.35 \\ 0.15 & 0.65 \end{bmatrix} \end{array} \begin{matrix} F \\ G \end{matrix} \quad \text{tomorrow} \]

On a particular Monday, 40% of coffee-buyers bought their coffees at Fatima's.

The matrix recursion relation \(S_{n+1} = T S_n\) is used to model this situation.

The percentage of coffee-buyers who are expected to buy their coffee at Giorgio's on Friday of the same week is closest to

• A. 31%
• B. 32%
• C. 34%
• D. 45%
• E. 68%

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Correct Answer: B
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The table below shows information about three matrices: \(A\), \(B\) and \(C\).

The transpose of matrix \(A\), for example, is written as \(A^T\).

What is the order of the product \(C^T \times (A^T \times B)^T\)?

• A. \(2 \times 3\)
• B. \(3 \times 4\)
• C. \(4 \times 2\)
• D. \(4 \times 3\)
• E. \(4 \times 4\)

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Correct Answer: E
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