https://file.aiquickdraw.com/mathbot/file/1726405684168t6cjinwz.jpg

The matrix provided is:

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

To find the eigenvalues and eigenvectors, we follow these steps:

### 1. **Find Eigenvalues:**
The eigenvalues \(\lambda\) are found by solving the characteristic equation:

\[
\det(A - \lambda I) = 0
\]

Where \(I\) is the identity matrix and \(\lambda\) is the eigenvalue.

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

Now, find the determinant:

\[
\det(A - \lambda I) = \det \begin{pmatrix} 2-\lambda & 1 & 0 \\ 0 & 2-\lambda & 1 \\ 0 & 0 & 2-\lambda \end{pmatrix}
\]

Since it's an upper triangular matrix, the determinant is simply the product of the diagonal elements:

\[
\det(A - \lambda I) = (2-\lambda)(2-\lambda)(2-\lambda) = (2-\lambda)^3
\]

Therefore, the eigenvalues are:

\[
\lambda_1 = \lambda_2 = \lambda_3 = 2
\]

### 2. **Find Eigenvectors:**
Now that we know the eigenvalue \(\lambda = 2\), we can find the eigenvectors by solving:

\[
(A - 2I)v = 0
\]

This simplifies to solving the system:

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

From the first row: \(x_2 = 0\).

From the second row: \(x_3 = 0\).

Thus, the eigenvectors take the form:

\[
v = \begin{pmatrix} x_1 \\ 0 \\ 0 \end{pmatrix}
\]

where \(x_1\) can be any non-zero value. Hence, the eigenvectors corresponding to the eigenvalue \(\lambda = 2\) are of the form:

\[
v = c \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}
\]

where \(c\) is any constant.

Would you like more detailed steps on the process? Here are 5 related questions to consider:

1. What is the geometric multiplicity of the eigenvalue \(\lambda = 2\)?
2. How do the properties of upper triangular matrices simplify finding eigenvalues?
3. Can a matrix with repeated eigenvalues still be diagonalizable?
4. How do you find a basis for the eigenspace corresponding to a specific eigenvalue?
5. What is the relationship between eigenvectors and the diagonalization of a matrix?

**Tip:** For upper triangular matrices, the eigenvalues are always the entries on the diagonal.

Find eigenvalues and eigenvectors of the given matrix.

Explore how to find the eigenvalues and eigenvectors of a 3x3 matrix, specifically focusing on cases where all eigenvalues are equal. The problem includes solving the characteristic equation and determining the corresponding eigenvectors for advanced linear algebra students.

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