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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!

What is the general solution of dY/dt = AY with A = [[-8, 2], [4, -1]]?

Explore how to find the general solution of the linear differential equation dY/dt = AY with matrix A = [[-8, 2], [4, -1]]. Learn about eigenvalues, eigenvectors, and the matrix exponential method for solving systems of first-order differential equations.

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