For each pair of matrices:
R_{ 5 \times 4} and S_{ 3 \times 4}
P_{ 4 \times 2} and Q_{ 2 \times 3}
B_{ 3 \times 2} and A_{ 5 \times 3}
R_{ 5 \times 3} and S_{ 4 \times 5}
Consider A=\begin{bmatrix} -7 & -3 & -1 \\ 5 & 7 & -6 \end{bmatrix}. If A is to be multiplied by B, a column matrix, state the dimensions of B.
For each pair of matrices:
State whether the product AB is defined. If yes, answer parts (ii) and (iii).
State the dimensions of AB.
Determine the matrix AB.
If B is a 5 \times 2 matrix and the product AB is a 4 \times 2 matrix, state the dimensions of A.
A matrix calculation of A= BC resulted in A=\begin{bmatrix} -4 & -7 & 7 \\ -1 & 6 & 2 \\ 8 & -6 & 9 \end{bmatrix}.
If B is a 3 \times 4 matrix, state the order of C.
The product BAC is a 3 \times 3 matrix where the matrix A = \begin{bmatrix} -4 & 2 & 1 \\ -3 & -1 & 4 \\ 5 & 9 & 7 \end{bmatrix}.
State the order of B.
State the order of C.
Consider A = \begin{bmatrix} -5 & -6 & 0 \\ 3 & 2 & -8 \end{bmatrix} \text{ and } B=\begin{bmatrix} 5 &-2 \\ -7 & -1 \\ 8 & 9 \end{bmatrix}.
State the dimensions of AB.
Determine the matrix AB.
State the dimensions of BA.
Determine the matrix BA.
Is AB equal to BA?
Solve the following matrix equations for n:
State the order of the resulting matrix after evaluating A \times A^{T}.
Evaluate the matrix multiplication, A\times A^{T}.
Determine whether the following matrices can be raised to a power:
For the following matrices, calculate A^2:
If A=\begin{bmatrix} -1 & 0 \\ 5 & -2 \end{bmatrix}, calculate:
A^{2}
A^{3}
If A=\begin{bmatrix} 3 & 1 \\ -1 & 2\end{bmatrix}, calculate:
A^{2}
A^{3}
A^{2} \times A
Does A^{3} equal A^{2} \times A?
Use your CAS calculator to answer the following. Round each element of the resulting matrix to three decimal places if required.
A=\begin{bmatrix} -5 & 3 \\ 2 & -1 \end{bmatrix}, calculate A^{7}.
A=\begin{bmatrix} 0.85 & 0.46 \\ 0.15 & 0.54\end{bmatrix}, calculate A^{27}.
A=\begin{bmatrix} -3 & 3 & 0 \\ -2 & 1 & 6 \\ 0 & 4 & 0\end{bmatrix}, calculate A^6.
.A=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}, calculate A^{16}.
A=\begin{bmatrix} 1 & -2 & 3 & 4 \\ 2 & 1 & -7 & 8 \\ 3 & 5 & 8 & -6 \\ -8 & -1 & 10 & 9 \end{bmatrix} , calculate A^{3}.
A=\begin{bmatrix} 0 & 0.4 & 0.7 & -0.1 \\ 0.9 & 0.8 & 0.7 & 0.2 \\ -0.1 & 0.5 & 0.5 & 0.1 \\ 0 & 0.2 & 0.2 & 0.6 \end{bmatrix} , calculate A^{15}.
A=\begin{bmatrix} 5 & 1 & 0 & 1 & 2 \\ 2 & 3 & 0 & 4 & 2 \\ 1 & 2 & 4 & 1 & 4 \\ 2 & 1 & 4 &3 & 0 \\ 0 & 5 & 2 & 1 & 2\end{bmatrix} , calculate A^{5}.
A=\begin{bmatrix} 0 & 0 & 0 & 0.1 & 0 \\ 0.4 & 0 & 0.2 & 0.8 & 0 \\ 0 & 0.5 & 0 & 0 & 0.6 \\ 0.3 & 0.3 & 0.3 & 0 & 0 \\ 0.2 & 0 & 0.6 & 0 & 0.4\end{bmatrix} , calculate A^{27}.
For each matrix, use your CAS calculator to answer the following, rounding each element of the solution to three decimal places:
Explain what will happen to the matrix A as it is raised to larger powers.
The map below shows four towns and the paths connecting them. Let matrix A represent:
A=\text{From}:\begin{matrix} \\ & \begin{matrix} \text{To: Kingston} & \text{Ashland} & \text{Greenville} & \text{Dunham} \end{matrix} \\ \begin{matrix} \text{Kingston} \\ \text{Ashland} \\ \text{Greenville} \\ \text{Dunham} \end{matrix} & \begin{bmatrix} 0 \qquad & \qquad 1 \qquad & \qquad 0 \qquad & \qquad 0 \\ 1 \qquad & \qquad 0 \qquad & \qquad 1 \qquad & \qquad 1 \\ 0 \qquad & \qquad 1 \qquad & \qquad 0 \qquad & \qquad 1 \\ 0 \qquad & \qquad 1 \qquad & \qquad 1 \qquad & \qquad 0 \end{bmatrix} \end{matrix}
Find A^{4}, the matrix that represents all possible four-step paths between the towns.
How many four-step paths can be taken from Greenville to Dunham?
Five friends are participating in a puzzle event. As part of the rules of the event, each person can only send messages to two other people. Let matrix A represent:
A= \text{Sender}:\begin{matrix} & \begin{matrix} \text{Receiver}: \\ \text{Aaron} & \text{Bill} & \text{Claudia} & \text{Darrel} & \text{Eve} \end{matrix} \\ \begin{matrix} \text{Aaron} \\ \text{Bill} \\ \text{Claudia} \\ \text{Darrel} \\ \text{Eve} \end{matrix} & \begin{bmatrix} 0 \qquad & \qquad 1 \qquad & \qquad 1 \qquad & 0 \qquad & 0 \\ 0 \qquad & \qquad0 \qquad & \qquad1 \qquad & 1\qquad & 0\\ 0 \qquad&\qquad 0\qquad & \qquad 0 \qquad & 1 \qquad & 1\\ 1 \qquad & \qquad 0 \qquad & \qquad 0 \qquad & 0 \qquad & 1\\ 1\qquad & \qquad1 \qquad&\qquad 0\qquad & 0 \qquad & 0 \end{bmatrix} \end{matrix}
Find A^{2}, the matrix that represents all messages that can be sent in two steps.
Determine if Bill can send a message to Aaron in a maximum of two steps.
A particular website is to be designed so that all website content is available to a user within three clicks. The webpage adjacency matrix A (representing available links between different parts of the website) is given.
If every webpage can be reached within three clicks, then the matrix B_3 = A + A^{2} + A^{3} will have no zero entries.
Find A^{2}.
Find A^{3}.
Find the matrix B_{3} = A + A^{2} + A^{3}.
Hence determine if every webpage is able to be reached within three clicks.
A local bakery was selling three different products yesterday. The tables show the price that each product was sold for and the amount sold:
\enspace
Price:
| Meat pie | Croissant | Bread roll |
|---|---|---|
| \$5 | \$3 | \$4 |
\enspace
Quantity sold:
| Meat pie | Croissant | Bread roll |
|---|---|---|
| 22 | 13 | 35 |
Organise the prices of each product into the row matrix in the order given in the table.
Organise the quantity sold of each product into the column matrix in the order given in the table.
Calculate the bakery's total revenue for the day by finding AB.
In the last cricket season, the Darwin Darts had 14 wins, 1 tie, 6 draws and 8 losses. The table shows the points system used in the competition:
Construct the team's results in each cricket match into a row matrix A. Order the elements of A from left to right as the number of wins, ties, draws and losses.
Construct the points system into a column matrix B. Order the elements of B from top to bottom as the points for each win, tie, draw and loss.
Calculate the team's total points for the season by finding AB.
| Result | Points |
|---|---|
| \text{Win} | 6 |
| \text{Tie} | 3 |
| \text{Draw} | 1 |
| \text{Loss} | 0 |
Frank owns two pizza stores, Panania Pizza and Penrith Pizza, at which he sells small pizzas for \$7, medium-sized pizzas for \$15 and large pizzas for \$28.
The table shows the number of pizzas sold at each store on a particular day:
| Small | Medium | Large | |
|---|---|---|---|
| Panania Pizza | 21 | 25 | 12 |
| Penrith Pizza | 26 | 11 | 22 |
Organise the prices into the column matrix A, in ascending size order.
Organise the number of pizzas sold into the matrix B, as given in the table.
Calculate Frank's total revenue for each store by finding BA.
Daniel's Dishes sells three different meals. The table shows the number of items they bundle together into each type of meal:
| Hamburgers | Onion Rings | Chicken nuggets | Pizzas | Soft drinks | |
|---|---|---|---|---|---|
| Mega Meal | 5 | 5 | 10 | 4 | 4 |
| Hungry Meal | 2 | 8 | 5 | 3 | 3 |
| Crazy Meal | 1 | 9 | 12 | 0 | 0 |
On Saturday, they sold 20 Mega Meals (M), 17 Hungry Meals (H) and 16 Crazy Meals (C). On Sunday, they sold 22 Mega Meals, 21 Hungry Meals and 25 Crazy Meals.
Organise the number of meals sold each day into the matrix A.
Organise the number of items in each meal into the matrix B as given in the table.
Find AB.
How many soft drinks did Daniel's Dishes sell on Saturday?
How many hamburgers did Daniel's Dishes sell on the weekend?