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4.03 Representing information using matrices

Worksheet
Introduction to matrices
1

What is the special property of a square matrix?

2

State whether the following matrices are valid:

a
\begin{bmatrix} 3 & 6 \\ -2 & 8 \end{bmatrix}
b
\begin{bmatrix} 3 & 6 \\ -2 & 5 & -4 \end{bmatrix}
3

How many elements does a matrix with 4 rows and 5 columns contain?

4

How many elements are in the third column of a 7 \times 8 matrix?

5

How many elements are in the leading diagonal of a 4 \times 4 matrix?

6

Suppose M is a 4 \times 3 matrix.

a

How many rows does M have?

b

How many columns does M have?

7

If a column matrix contains 6 elements, how many rows does the matrix have?

8

How many columns does the identity matrix I_4 contain?

9

Write down the dimensions of the following matrices:

a
\begin{bmatrix} 1 & 4 \\ 8 & -3 \\ 9 & 0 \end{bmatrix}
b
\begin{bmatrix} 3 & -5 & 11 & 16 & -8 \end{bmatrix}
c
\begin{bmatrix} 7 \\ -3 \\ -2 \\ 6 \end{bmatrix}
d
\begin{bmatrix} 1 & 4 \\ -8 & 5 \\ -2 & 7 \\ 0 & 12 \end{bmatrix}
e
\begin{bmatrix} 1 & 4 & -8 \\ 11 & 17 & -15 \\ 21 & 10 & 9 \end{bmatrix}
f
\begin{bmatrix} 6 & 8 & 3 & -3 & 1 \\ 7 & 3 & 10 & 2 & 8 \\ 9 & 1 & -2 & 8 & 7 \\ 8 & -5 & 3 & 2 & 9 \end{bmatrix}
g
\begin{bmatrix} 8 & -1 & -4 & 9\\ 12 & 15 & -11 & 10 \end{bmatrix}
10

If a matrix has 10 elements, how many different dimensions could it possibly have?

11

A matrix with three less rows than columns has 54 elements. What are the dimensions of this matrix?

12

Write down the entry at a_{32} in the matrix \begin{bmatrix} 7 & 3 & -4 & 10\\ 12 & 0 & -11 & 10 \\ 8 & -2 & 9 & 11 \end{bmatrix}

13

Where is 4 located in the matrix A = \begin{bmatrix} 9 & 2 & -6 & 8\\ 3 & 2 & 0 & -1 \\ 9 & 5 & 4 & 11 \end{bmatrix}

14

Consider the matrices A = \begin{bmatrix} 4 & 1 & -2\\ 3 & 5 & 0 \end{bmatrix} and B = \begin{bmatrix} 6 & 2 \\ 3 & 0 \\ -1 & 9 \end{bmatrix}.

a

Find the value of a_{21} - b_{11}

b

Find the value of a_{13} \times b_{31}

15

Consider the following matrices:

A = \begin{bmatrix} 9 & 2 & 8 \\ 1 & 10 & 8 \\ 8 & -3 & 5 \end{bmatrix}, B = \begin{bmatrix} 9 & 2 & -6 & 8 & 10 \end{bmatrix}, C = \begin{bmatrix} 10 \\ 8 \\ -9 \\ 1 \end{bmatrix}, and D = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Which of these matrices is a:

a

Row matrix

b

Column matrix

c

Square matrix

d

Zero matrix

16

Write down the 3 \times 3 identity matrix.

Organising data into matrices
17

The following table shows the number of customers walking through the doors of two businesses over the long weekend:

a

Organise the data into a matrix. Let the position of each element be the same as the position in the table.

b

What are the dimensions of the matrix?

SaturdaySundayMonday
Florist14011098
Butcher150118135
18

The table shows the average temperature (in degrees Celsius) in two cities for each of the four seasons:

SummerAutumnWinterSpring
Perth30251728
Melbourne32261527

This data is organised into the following matrix:

\begin{bmatrix} 30 & 32 \\ 25 & 26 \\ 17 & 15 \\ 28 & 27 \end{bmatrix}
a

Do the rows of the matrix represent the temperatures for each season or for each city?

b

What are the dimensions of the matrix?

19

The following are the costs of a train ticket during different time periods:

  • Weekdays: \$6 peak, \$3 off-peak

  • Weekends: \$10 peak, \$7 off-peak

  • Public Holidays: \$16 peak, \$12 off-peak

a

Organise the data into a 3 \times 2 matrix.

b

Organise the data into a 2 \times 3 matrix.

20

Jack, a chef, is known for the following two recipes:

Crazy Cookie which contains:

  • 360 g of yeast

  • 410 g of salt

  • 340 g of flour

  • 230 g of sugar

  • 120 g of honey

Scrumptious Surprise which contains:

  • 420 g of yeast

  • 390 g of salt

  • 330 g of flour

  • 200 g of sugar

  • 80 g of honey

Organise the amounts into a 2 \times 5 matrix.

21

A long jump competition was down to four competitors in the final. Uther jumped 7.4m, 5.7 \text{ m} and 7.5 \text{ m} in his attempts. Yuri jumped 6.7 \text{ m}, 7.3 \text{ m} and 6.9 \text{ m}. Vincent jumped 7.1 \text{ m}, 5.9 \text{ m} and 6.8 \text{ m}, and Luigi jumped 5.6 \text{ m}, 6.1 \text{ m} and 6.3 \text{ m}.

Organise the data into a 3 \times 4 matrix.

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Outcomes

1.2.4

use matrices for storing and displaying information that can be presented in rows and columns; for example, databases, links in social or road networks

1.2.5

recognise different types of matrices (row, column, square, zero, identity) and determine their size

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