The matrix


FROM
R D I
[ 0.6 0.1 0.1] R
P=[ 0.2 0.7 0.1] D TO
[ 0.2 0.2 0.8] I


To
is called a stochastic matrix. Each entry
pij (i ≠ j)
represents the proportion of the voting population that changes from Party i to Party j, and
pii
represents the proportion that remains loyal to the party from one election to the next. Use a graphing utility to find P3, P4, P5, P6, P7, and P8 for the matrix given. (Round your answers to four decimal places.)
P3=
P4=
P5=
P6=
P7=
P8=

Please can somebody tell me how I van develop this one? Thank you!!

To find P3, P4, P5, P6, P7, and P8 for the given matrix, we need to raise the matrix P to the power of 3, 4, 5, 6, 7, and 8, respectively. Each power represents the number of elections or transitions from one party to another.

To calculate the result, we can use a graphing utility or matrix calculation software. Let's use matrix calculation software like Python, MATLAB, or Excel to find the results.

Here's an example of how you can solve this using Python:

```python
import numpy as np

# Define the matrix P
P = np.array([[0.6, 0.1, 0.1],
[0.2, 0.7, 0.1],
[0.2, 0.2, 0.8]])

# Calculate P3, P4, P5, P6, P7, and P8
P3 = np.linalg.matrix_power(P, 3)
P4 = np.linalg.matrix_power(P, 4)
P5 = np.linalg.matrix_power(P, 5)
P6 = np.linalg.matrix_power(P, 6)
P7 = np.linalg.matrix_power(P, 7)
P8 = np.linalg.matrix_power(P, 8)

# Round the results to four decimal places
P3 = np.round(P3, 4)
P4 = np.round(P4, 4)
P5 = np.round(P5, 4)
P6 = np.round(P6, 4)
P7 = np.round(P7, 4)
P8 = np.round(P8, 4)

# Print the results
print("P3 =")
print(P3)
print("P4 =")
print(P4)
print("P5 =")
print(P5)
print("P6 =")
print(P6)
print("P7 =")
print(P7)
print("P8 =")
print(P8)
```

This code uses the numpy library to perform matrix calculations. It defines the matrix P based on the given values. Then it uses the `np.linalg.matrix_power()` function to calculate P3, P4, P5, P6, P7, and P8. Finally, it rounds the results to four decimal places and prints them.

You can run this code in any Python environment, such as Anaconda, Jupyter Notebook, or an IDE like PyCharm. Just make sure you have the numpy library installed.