14.01 Representing information using matrices
What is the special property of a square matrix?
State whether the following matrices are valid:
How many elements does a matrix with 4 rows and 5 columns contain?
How many elements are in the third column of a 7 \times 8 matrix?
How many elements are in the leading diagonal of a 4 \times 4 matrix?
Suppose M is a 4 \times 3 matrix.
How many rows does M have?
How many columns does M have?
If a column matrix contains 6 elements, how many rows does the matrix have?
How many columns does the identity matrix I_4 contain?
Write down the dimensions of the following matrices:
If a matrix has 10 elements, how many different dimensions could it possibly have?
A matrix with three less rows than columns has 54 elements. What are the dimensions of this matrix?
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}
Where is 4 located in the matrix A = \begin{bmatrix} 9 & 2 & -6 & 8\\ 3 & 2 & 0 & -1 \\ 9 & 5 & 4 & 11 \end{bmatrix}
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}.
Find the value of a_{21} - b_{11}
Find the value of a_{13} \times b_{31}
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:
Row matrix
Column matrix
Square matrix
Zero matrix
Write down the 3 \times 3 identity matrix.
The following table shows the number of customers walking through the doors of two businesses over the long weekend:
Organise the data into a matrix. Let the position of each element be the same as the position in the table.
What are the dimensions of the matrix?
| Saturday | Sunday | Monday | |
|---|---|---|---|
| Florist | 140 | 110 | 98 |
| Butcher | 150 | 118 | 135 |
The table shows the average temperature (in degrees Celsius) in two cities for each of the four seasons:
| Summer | Autumn | Winter | Spring | |
|---|---|---|---|---|
| Perth | 30 | 25 | 17 | 28 |
| Melbourne | 32 | 26 | 15 | 27 |
This data is organised into the following matrix:
Do the rows of the matrix represent the temperatures for each season or for each city?
What are the dimensions of the matrix?
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
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Public Holidays: \$16 peak, \$12 off-peak
Organise the data into a 3 \times 2 matrix.
Organise the data into a 2 \times 3 matrix.
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.
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.