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The problem asks to find the vector \(\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\) that satisfies the equation \(A\mathbf{x} = 3\mathbf{x}\), where the matrix \(A\) is:

\[
A = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 3 & 1 \\ 1 & 0 & 2 \end{pmatrix}
\]

This is a classic eigenvalue problem where we are given the matrix \(A\) and the scalar 3 (which is the eigenvalue), and we need to find the corresponding eigenvector \(\mathbf{x}\).

To solve this, we use the equation for eigenvectors:

\[
A\mathbf{x} = \lambda \mathbf{x}
\]

In this case, \(\lambda = 3\), so the equation becomes:

\[
A\mathbf{x} = 3\mathbf{x}
\]

This can be rewritten as:

\[
(A - 3I)\mathbf{x} = 0
\]

Where \(I\) is the identity matrix, and \(A - 3I\) is:

\[
A - 3I = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 3 & 1 \\ 1 & 0 & 2 \end{pmatrix} - \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix} = \begin{pmatrix} -2 & 2 & 1 \\ 0 & 0 & 1 \\ 1 & 0 & -1 \end{pmatrix}
\]

Now, solve the system \((A - 3I)\mathbf{x} = 0\), which is equivalent to solving the following system of equations:

\[
\begin{pmatrix} -2 & 2 & 1 \\ 0 & 0 & 1 \\ 1 & 0 & -1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
\]

I will solve this system for \(\mathbf{x}\).The eigenvector corresponding to the eigenvalue \( \lambda = 3 \) is approximately:

\[
\mathbf{x} = \begin{pmatrix} 1.94 \\ 1.54 \\ 1 \end{pmatrix}
\]

This is a solution to the equation \(A\mathbf{x} = 3\mathbf{x}\), where \(A\) is the given matrix.

Would you like more details on the steps taken or further clarification on the eigenvalue problem?

Here are 5 questions related to this concept:
1. What are eigenvalues and eigenvectors in linear algebra?
2. How do you solve systems of linear equations like the one presented?
3. What is the significance of the eigenvalue 3 in this problem?
4. How can you compute eigenvalues and eigenvectors for larger matrices?
5. Why is normalizing eigenvectors useful in these problems?

**Tip:** In eigenvalue problems, the eigenvector can be scaled by any non-zero constant, as it represents a direction, not a specific magnitude.

Given the 3x3 matrix A = [[1, 2, 1], [0, 3, 1], [1, 0, 2]], find the vector x = (x1, x2, x3) that satisfies the equation Ax = 3x.

Learn how to find the eigenvector for the given 3x3 matrix A = [[1, 2, 1], [0, 3, 1], [1, 0, 2]] with an eigenvalue of 3. This problem involves concepts of linear algebra and eigenvalues, suitable for undergraduate students studying linear algebra.

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