Calculate the following for A=\begin{bmatrix} 5 & 7 \\ 8 & 6 \end{bmatrix}:
For the matrices A, B and C, calculate the following. Round matrix elements to four decimal places.
A=\begin{bmatrix} 0.8 & 0.6 \\ 0.2 & 0.4 \end{bmatrix},B=\begin{bmatrix} 0.83 & 0.57 \\ 0.17 & 0.43 \end{bmatrix},C=\begin{bmatrix} 0.1 & 0.3 & 0.6 \\ 0.6 & 0.1 & 0.3 \\ 0.3 & 0.6 & 0.1 \end{bmatrix}Simplify and evaluate the expression A^{2} + 3 B - C^{3}, where A, B and C are defined below:
A=\begin{bmatrix} 9 & 7 \\ 1 & 8 \end{bmatrix},B=\begin{bmatrix} 2 & 5 \\ 9 & 7 \end{bmatrix},C=\begin{bmatrix} 8 & 2 \\ 4 & 4 \end{bmatrix}
Construct the transition matrix T for each of the following state diagrams:
The chance of the weather being rainy (R) or fine (F) tomorrow, given it was rainy or fine today, is displayed in the table below. Construct an appropriate transition matrix for this information.
| \text{Today} (R) | \text{Today} (F) | |
|---|---|---|
| \text{Tomorrow} (R) | 0.8 | 0.25 |
| \text{Tomorrow} (F) | 0.2 | 0.75 |
Consider the transition matrix T below:
T=\begin{bmatrix} 0.758 & 0.153 \\ 0.243 & 0.847 \end{bmatrix}
Find T^{50}, rounding each element correct to four decimal places:
Find T^{51}, rounding each element correct to four decimal places:
Hence, determine whether a steady state been reached.
Consider the transition matrix T below:
T=\begin{bmatrix} 0.07 & 0.03 & 0.95 \\ 0.04 & 0.20 & 0 \\ 0.88 & 0.77 & 0.05 \end{bmatrix}
Find T^{20}, rounding each element correct to four decimal places.
Find T^{21}, rounding each element correct to four decimal places.
Hence, determine whether a steady state been reached.
Consider the transition matrix T given below:
T=\begin{bmatrix} 0 & 0.19 & 0 & 0.58 \\ 0.67 & 0 & 0.45 & 0 \\ 0 & 0.81 & 0 & 0.42 \\ 0.33 & 0 & 0.55 & 0 \end{bmatrix}
Calculate the matrix T^{2}.
Explain why this matrix has no steady state solution.
Each summer the families of children in a school either stay at home (H) or go away on vacation (V). The activities for the holidays change according to the transition matrix below:
If 100 families stay at home in 2016, how many of these 100 families are expected to stay at home in 2017?
How many of these 100 families are expected to go on vacation in 2017?
How many of the families who stay home in 2017 are expected to stay at home in 2018 also?
A factory has 3 warehouses A, B and C. Forklift trucks are used to transport goods between the 3 warehouses. They start the day in one warehouse and end the day at the same or a different warehouse. The matrix, T, represents the transition matrix for this situation:
Calculate the percentage of forklift trucks which start the day at factory A and end the day at factory A.
There were 50 forklift trucks that started the day at factory B. Determine the number of forklift trucks expected to be parked at factory C at the end of the day. Round your answer to the nearest whole number.
A website uploads two blog posts each day, one about social issues \left ( S \right ) and the other about environmental concerns \left ( E \right ).
Of the people who read a blog every day, 52\% of those that read about social issues on one day will read about social issues the next day. Also, 83\% of those that read about environmental concerns will also read about environmental concerns the next day. Construct a state diagram that best represents this information.
Construct the transition matrix T that represents the transitional probabilities between each state.
On a certain day, the website records that 800 people read blog \left( E \right) while 550 people read \left( S \right). Use this information to predict the number of readers that will read blog \left( E \right) in 3 days time. Round your answer to the nearest whole number.
Students at an after school care program have two choices of activities to do in their free time, read a book (B) or play with an abacus (A). It was found that 25\% of children who read books one afternoon will play with the abacus the next afternoon, and 30\% of children who play with the abacus one afternoon will read books the next afternoon.
Construct the transition matrix for the situation outlined above.
If in one afternoon, 20 students play with the abacus and another 30 students read a book, how many will be expected to play with the abacus the next afternoon? Round your answer to the nearest whole number.
Explain why this model is unrealistic.
Irene (I) and Larry (L) are playing chess. They each have an equal chance of winning the first game. If Irene wins, then she gains confidence and her chance of winning the next game becomes 70\%. If Larry wins, his chance of winning the next game is 60\%.
Construct the initial probability matrix P_0 for the situation outlined above.
Construct the transition matrix T, which displays the probability of each player winning, given each player has already won.
Calculate P_1, the probabilty matrix after one game has been played.
Calculate the probability that Larry wins the second game.
Determine P_5, the probability matrix after five games have been played.
In the year 2019, there were 220\,000 people living in Town A and 60\,000 people in Town B. Each year, 5\% of the people in Town A move to Town B, and 23\% of the people in Town B move to Town A.
Construct the initial population matrix P_0.
Construct the transition matrix T that represents the shift in population.
Find the prediction of the population in each town in 2020.
Find the prediction of the population in each town in 2023.
Explain why this model is unrealistic.
Each year two stores, Supplies R Us (S) and The General (G) compete for business during the holiday period. They used surveys as a way of identifying how successful their marketing campaigns are. The surveys showed that 75\% of customers will return to Supplies R Us if they purchased goods there in the previous year, and 70\% of The General's customers will return to them year to year.
Construct a transition matrix T to represent the change in customers each year.
If both stores have 600 customers initially, find S_2, the matrix containing the expected number of customers that return to each store after two years.
Find S_7, the matrix containing the expected number of customers that return to each store after seven years.
A computer system can operate in two different modes. Every hour, it either remains in the same mode or switches to the other mode, according to the following initial transition probability matrix P:
Compute the two-step transition probability matrix, P^{2}.
If the system is in Mode 1 at 3:00 pm calculate the probability that it will be in Mode 1 at 8:00 pm. Round your answer to three decimal places.