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.