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To find the general solution to this system of differential equations, we have to solve:

\[
\mathbf{x}'(t) = A \mathbf{x}(t),
\]

where 
\[
A = \begin{pmatrix} 1 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 1 \end{pmatrix}.
\]

The general approach involves the following steps:

### Step 1: Find the Eigenvalues of \( A \)
1. Compute the characteristic polynomial of \( A \) by solving \( \det(A - \lambda I) = 0 \).
2. This will provide the eigenvalues \( \lambda_1 \), \( \lambda_2 \), and \( \lambda_3 \).

### Step 2: Find the Eigenvectors
1. For each eigenvalue \( \lambda_i \), solve \( (A - \lambda_i I) \mathbf{v} = 0 \) to find the corresponding eigenvector \( \mathbf{v}_i \).

### Step 3: Construct the General Solution
1. Using the eigenvalues and eigenvectors, the general solution to the system will take the form:
   \[
   \mathbf{x}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2 + c_3 e^{\lambda_3 t} \mathbf{v}_3,
   \]
   where \( c_1 \), \( c_2 \), and \( c_3 \) are constants determined by initial conditions.

Let me carry out these steps for you.The matrix \( A \) has three distinct eigenvalues and corresponding eigenvectors as follows:

1. **Eigenvalue** \( \lambda_1 = -1 \) with eigenvector \( \mathbf{v}_1 = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} \).
2. **Eigenvalue** \( \lambda_2 = 1 \) with eigenvector \( \mathbf{v}_2 = \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} \).
3. **Eigenvalue** \( \lambda_3 = 4 \) with eigenvector \( \mathbf{v}_3 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \).

### General Solution
The general solution to the system is given by:

\[
\mathbf{x}(t) = c_1 e^{-t} \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} + c_2 e^{t} \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} + c_3 e^{4t} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},
\]

where \( c_1 \), \( c_2 \), and \( c_3 \) are constants determined by initial conditions.

Would you like further details on any part of this solution?

Here are five related questions you might find helpful:
1. How do we verify that these eigenvalues and eigenvectors satisfy the original matrix equation?
2. What if the matrix \( A \) had repeated eigenvalues? How would the solution change?
3. Can we solve this system if we were given initial conditions?
4. How does the behavior of the solution change as \( t \to \infty \)?
5. How would we approach this problem if \( A \) were a non-diagonalizable matrix?

**Tip:** Understanding how eigenvalues affect the solution’s growth or decay is crucial, especially when analyzing the stability of systems.

Find the general solution to the equation:
\[
\mathbf{x}'(t) = A \mathbf{x}(t),
\]
where
\[
A = \begin{pmatrix} 1 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 1 \end{pmatrix}.
\]

Find the general solution to a system of differential equations involving a 3x3 matrix. Learn how to calculate eigenvalues, eigenvectors, and construct the solution. This solution uses linear algebra techniques suitable for university-level students.

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