To determine the second row of the matrix $A$, let's first note what it means for a matrix to be stochastic:

1. Definition of a Stochastic Matrix: Each row of a stochastic matrix must sum to 1, and all entries must be non-negative.

The given matrix is: $A = \begin{bmatrix} 0.6 & 0.5 \\ ? & ? \end{bmatrix}$

For $A$ to be a stochastic matrix:

• The sum of the elements in each row must equal 1.

Step 1: Find the second row.

Let the second row be $[a, b]$. Then: $a + b = 1$

The image indicates that the second row is given as $[0.4, 0.6]$, so: $0.4 + 0.6 = 1$ which satisfies the requirement for a stochastic matrix.

Step 2: Verify the steady state vector.

The steady state vector for $A$ is given as: $\begin{bmatrix} \frac{5}{9} \\ \frac{4}{9} \end{bmatrix}$ This vector satisfies $A \cdot v = v$, indicating that it is indeed the steady state.

If you would like further details on why this vector is the steady state or have other questions, let me know.

Here are five questions related to the topic:

1. How do you compute the steady state vector of a stochastic matrix?
2. What properties must a matrix have to be considered a Markov matrix?
3. How would the steady state vector change if the matrix was modified?
4. What are the applications of steady state vectors in real-world problems?
5. How can one determine the convergence rate of $A^n \cdot v$ as $n \to \infty$?

Tip: To find the steady state vector for a stochastic matrix, solve the equation $A \cdot v = v$, where $v$ is a probability vector (its components sum to 1).