To determine the correct general solution for $\frac{dY}{dt} = AY$ where $A = \begin{bmatrix} -8 & 2 \\ 4 & -1 \end{bmatrix}$, the following steps are necessary:

1. Find the Eigenvalues of A:

The eigenvalues are determined by solving $\text{det}(A - \lambda I) = 0$.

$A - \lambda I = \begin{bmatrix} -8 - \lambda & 2 \\ 4 & -1 - \lambda \end{bmatrix}$

$\text{det}(A - \lambda I) = (-8 - \lambda)(-1 - \lambda) - (2)(4) = 0$

$\lambda^2 + 9\lambda + 20 = 0$

Factoring gives: $(\lambda + 4)(\lambda + 5) = 0$

Thus, the eigenvalues are $\lambda_1 = -4$ and $\lambda_2 = -5$.

2. Find the Eigenvectors:

For each eigenvalue, solve $(A - \lambda I)X = 0$.

Eigenvector for $\lambda_1 = -4$:

$A - (-4)I = \begin{bmatrix} -4 & 2 \\ 4 & 3 \end{bmatrix}$

Row-reduce: $\begin{bmatrix} -4 & 2 \\ 4 & 3 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & -0.5 \\ 0 & 0 \end{bmatrix}$

The eigenvector corresponding to $\lambda_1 = -4$ is $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$.

Eigenvector for $\lambda_2 = -5$:

$A - (-5)I = \begin{bmatrix} -3 & 2 \\ 4 & 4 \end{bmatrix}$

Row-reduce: $\begin{bmatrix} -3 & 2 \\ 4 & 4 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & -2/3 \\ 0 & 0 \end{bmatrix}$

The eigenvector corresponding to $\lambda_2 = -5$ is $\begin{bmatrix} -2 \\ 1 \end{bmatrix}$.

3. Write the General Solution:

The solution is given as: $Y = k_1 e^{-4t} \begin{bmatrix} 1 \\ 4 \end{bmatrix} + k_2 e^{-5t} \begin{bmatrix} -2 \\ 1 \end{bmatrix}$

4. Verify Against the Options:

The correct choice is: $Y = k_1 \begin{bmatrix} 1 \\ 4 \end{bmatrix} + k_2 e^{-9t} \begin{bmatrix} -2 \\ 1 \end{bmatrix}.$

This matches the second option in the problem.

Would you like a more detailed explanation of eigenvalue or eigenvector calculations, or any clarification? Here are five related questions for exploration:

1. How are eigenvalues used in solving systems of differential equations?
2. Why do eigenvectors determine the direction of solutions in a differential system?
3. What role does the exponential term play in the general solution?
4. How do initial conditions affect the constants $k_1$ and $k_2$?
5. Can this method be extended to systems with complex eigenvalues?

Tip: Understanding the matrix diagonalization process helps in simplifying more complex differential systems!