Calculate the following for A=\begin{bmatrix} 5 & 7 \\ 8 & 6 \end{bmatrix}:
For the transition 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}\vert A \vert
\vert B \vert
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}
The transition matrix, T, has a determinant of \vert T \vert =0.4. Find \vert T^3 \vert .
The transition matrix, T, has a determinant of \vert T \vert =-0.5. Find \vert T^2 \vert .
A matrix, A, has \vert A \vert = 0.3 and \vert A^4 \vert = 0.0072. Is A a transition matrix? Explain your answer.
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 below:
T=\begin{bmatrix} 0.4 & 0.3 & 0.1 \\ 0.5 & 0 & 0.6 \\ 0.1 & 0.7 & 0.3 \end{bmatrix}
Determine the smallest value of n such that T^{n} has reached a steady state.
Determine whether the following transition matrices will reach a steady state:
T=\begin{bmatrix} 0 & 0.91 & 0 & 0.49 \\ 0.2 & 0 & 0.29 & 0 \\ 0 & 0.09 & 0 & 0.51 \\ 0.8 & 0 & 0.71 & 0 \end{bmatrix}
T=\begin{bmatrix} 0.38 & 0 & 0.34 \\ 0.29 & 0.42 & 0.16 \\ 0.33 & 0.58 & 0.5 \end{bmatrix}
T=\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}
T=\begin{bmatrix} 0.5 & 0.7 \\ 0.5 & 0.3 \end{bmatrix}
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.
Consider the transition matrix T and initial state matrix S_0 below:
T=\begin{bmatrix} 0.5 & 0.2 \\ 0.5 & 0.8 \end{bmatrix},S_0=\begin{bmatrix} 160 \\ 220 \end{bmatrix}
Use the recurrence relation S_{n + 1} = T \times S_n to determine S_3.
Use S_n = T^n \times S_0 to calculate S_5.
Find the steady state solution matrix with each element rounded to two decimal places.
For each transition matrix T and initial state matrix S_0 below, determine the steady state solution matrix for the system, rounding each element to the nearest whole number if necessary.