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To solve this problem, let's go through each part step-by-step.

### 1. Find the Second Row of Matrix \( A \):
- **Matrix \( A \) is given as:**

  \[
  A = \begin{pmatrix} 0.4 & 0.5 \\ ? & ? \end{pmatrix}
  \]

- **Given that \( A \) is a stochastic matrix**, each row of a stochastic matrix sums to 1. For the first row, we have:

  \[
  0.4 + 0.5 = 0.9
  \]

  So, the missing element in the first row must be \( 0.1 \) to sum to 1.

- For the second row, let the unknown values be \( a \) and \( b \). The condition that the sum is 1 means:

  \[
  a + b = 1
  \]

  However, there isn't enough information in the problem to find the exact values of \( a \) and \( b \). But if we assume symmetry (which is common in basic examples of stochastic matrices), then we could set \( a = 0.5 \) and \( b = 0.5 \).

  Thus, the second row is:

  \[
  \text{Second row: } 0.5 \quad 0.5
  \]

### 2. Find the Steady State Vector for \( A \):
- **To find the steady-state vector \( x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \) such that \( A \cdot x = x \), solve the following equations:**

  \[
  0.4x_1 + 0.5x_2 = x_1
  \]
  \[
  0.5x_1 + 0.5x_2 = x_2
  \]

- **Reorganize the equations**:

  \[
  -0.6x_1 + 0.5x_2 = 0
  \]
  \[
  0.5x_1 - 0.5x_2 = 0
  \]

  From the second equation:

  \[
  x_1 = x_2
  \]

  Substitute \( x_1 = x_2 \) into the first equation:

  \[
  -0.6x_1 + 0.5x_1 = 0
  \]
  \[
  -0.1x_1 = 0
  \]

  The steady-state vector satisfies \( x_1 + x_2 = 1 \). Given that \( x_1 = x_2 \):

  \[
  x_1 = 0.5, \, x_2 = 0.5
  \]

  Thus, the steady state vector is:

  \[
  \text{Steady state vector: } \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix}
  \]

### 3. Find the Limit of \( A^n v \):
- **Given \( v = \begin{pmatrix} 16/7 \\ 7/7 \end{pmatrix} \):**

  As \( n \to \infty \), \( A^n \) applied to any initial vector \( v \) will approach the steady-state vector of \( A \).

  So, \( A^n v \) approaches:

  \[
  \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix}
  \]

---

### Final Answers:
1. Second row: \( 0.5, \, 0.5 \)
2. Steady state vector: \( \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix} \)
3. \( A^n v \) approaches: \( \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix} \)

Would you like more details on any of these steps? Here are some related questions:

1. What is a stochastic matrix and why do its rows sum to 1?
2. How can you solve for the steady-state vector of a matrix?
3. What is the significance of the steady-state vector in Markov chains?
4. How does the structure of a stochastic matrix affect its steady-state behavior?
5. What methods can be used to find the limit of \( A^n \) as \( n \) approaches infinity?

**Tip**: In a stochastic matrix, each column or row represents transition probabilities, and they must sum to 1 to represent a complete probability distribution.

What is the second row of the matrix A, if A is stochastic? A = [0.4 0.5 ? ?], The steady state vector for A is ___. If v = [16/7 7/7], then A^n v approaches ___ as n gets large.

This problem involves determining the second row of a stochastic matrix, finding its steady-state vector, and analyzing the long-term behavior of matrix powers applied to an initial vector. Learn how to apply properties of stochastic matrices and steady-state vectors to solve the problem.

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