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.