Questions: - You have learned how matrices model changing wetlands and the spread of a disease. What are some other contexts where matrices would be useful?
- How do you recognize a Markov matrix?
Transcript text: - You have learned how matrices model changing wetlands and the spread of a disease. What are some other contexts where matrices would be useful?
- How do you recognize a Markov matrix?
Solution
Solution Steps
Matrices are useful in various contexts such as computer graphics (for transformations and rotations), economics (for input-output models), and network theory (for representing graphs and connectivity).
A Markov matrix, also known as a stochastic matrix, is recognized by having all its entries be non-negative and each column summing to 1.
Step 1: Define the Matrix
We are given the matrix:
[0.50.50.20.80.30.7]
Step 2: Check Non-Negativity
We need to verify that all entries in the matrix are non-negative:
0.5≥0,0.2≥0,0.3≥0,0.5≥0,0.8≥0,0.7≥0
Since all entries are non-negative, this condition is satisfied.
Step 3: Check Column Sums
We need to check that the sum of each column is 1:
0.5+0.5=1,0.2+0.8=1,0.3+0.7=1
Since the sum of each column is 1, this condition is also satisfied.
Final Answer
The given matrix is a Markov matrix because it satisfies both conditions: all entries are non-negative, and the sum of each column is 1.