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The problem requires finding the eigenvalues of the matrix expression \( 3A^3 + 5A^2 - 6A + 2I \), where \( A \) is a given 3x3 matrix:

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

We will approach this step-by-step:

1. **Step 1: Eigenvalues of matrix \( A \)**:
   First, we need to compute the eigenvalues of matrix \( A \), as they will help simplify the calculation of the given matrix expression.

2. **Step 2: Use eigenvalues of \( A \)**:
   If \( \lambda \) is an eigenvalue of \( A \), then for the matrix expression \( 3A^3 + 5A^2 - 6A + 2I \), the corresponding eigenvalue will be:

   \[
   3\lambda^3 + 5\lambda^2 - 6\lambda + 2
   \]

Let me first calculate the eigenvalues of matrix \( A \). I'll proceed with this calculation now.The eigenvalues of matrix \( A \) are \( \lambda_1 = 1 \), \( \lambda_2 = 3 \), and \( \lambda_3 = 2 \).

Next, for each eigenvalue \( \lambda \), we compute the corresponding eigenvalue of the matrix expression \( 3A^3 + 5A^2 - 6A + 2I \) using the formula:

\[
3\lambda^3 + 5\lambda^2 - 6\lambda + 2
\]

I'll now calculate this for \( \lambda_1 = 1 \), \( \lambda_2 = 3 \), and \( \lambda_3 = 2 \).The eigenvalues of the matrix expression \( 3A^3 + 5A^2 - 6A + 2I \) are:

- For \( \lambda_1 = 1 \): eigenvalue = \( 4 \)
- For \( \lambda_2 = 3 \): eigenvalue = \( 110 \)
- For \( \lambda_3 = 2 \): eigenvalue = \( 34 \)

These are the eigenvalues of the given matrix expression.

Would you like further clarification or any additional details?

Here are 5 related questions to explore further:
1. How are eigenvalues related to matrix transformations?
2. Can we find eigenvectors associated with these eigenvalues?
3. What is the significance of the identity matrix in this context?
4. How does matrix exponentiation (like \( A^2 \) and \( A^3 \)) affect its eigenvalues?
5. How can we use the characteristic polynomial to directly find eigenvalues?

**Tip**: Eigenvalues provide insights into the stability and behavior of a system, especially in linear transformations and differential equations.

Find the eigenvalues of 3A^3 + 5A^2 - 6A + 2I where A = [[1, 2, -3], [0, 3, 2], [0, 0, 2]]

Find the eigenvalues of the matrix expression 3A^3 + 5A^2 - 6A + 2I where A is a 3x3 matrix. This solution covers linear algebra concepts including eigenvalues, matrix polynomials, and matrix powers, suitable for undergraduate-level students.

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