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?

- 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: \[ \begin{bmatrix} 0.5 & 0.2 & 0.3 \\ 0.5 & 0.8 & 0.7 \end{bmatrix} \]

Step 2: Check Non-Negativity

We need to verify that all entries in the matrix are non-negative: \[ 0.5 \geq 0, \quad 0.2 \geq 0, \quad 0.3 \geq 0, \quad 0.5 \geq 0, \quad 0.8 \geq 0, \quad 0.7 \geq 0 \] 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: \[ \begin{aligned} &0.5 + 0.5 = 1, \\ &0.2 + 0.8 = 1, \\ &0.3 + 0.7 = 1 \end{aligned} \] Since the sum of each column is 1, this condition is also satisfied.