For the matrix A= [(1 2 -3)( 0 3 2 )(0 0 -2)] find the eigen values of 3A³+5A²-6A+2I ​

We are tasked with finding the eigenvalues of the matrix expression \( 3A^3 + 5A^2 - 6A + 2I \) for the given matrix \( A \). 

### Step 1: Identify the Matrix \( A \)
Given matrix \( A \):
\[
A = \begin{pmatrix}
1 & 2 & -3 \\
0 & 3 & 2 \\
0 & 0 & -2
\end{pmatrix}
\]

### Step 2: Eigenvalues of Matrix \( A \)
The first step in solving this problem is to find the eigenvalues of the matrix \( A \). Since \( A \) is a triangular matrix (upper triangular in this case), the eigenvalues are simply the entries on the main diagonal. Therefore, the eigenvalues of \( A \) are:
\[
\lambda_1 = 1, \quad \lambda_2 = 3, \quad \lambda_3 = -2
\]

### Step 3: Eigenvalues of the Expression \( 3A^3 + 5A^2 - 6A + 2I \)
If \( \lambda \) is an eigenvalue of \( A \), then the corresponding eigenvalue of the expression \( 3A^3 + 5A^2 - 6A + 2I \) is obtained by substituting \( \lambda \) into the same expression.

For an eigenvalue \( \lambda \) of \( A \), the corresponding eigenvalue of the matrix expression \( 3A^3 + 5A^2 - 6A + 2I \) is:
\[
\mu = 3\lambda^3 + 5\lambda^2 - 6\lambda + 2
\]

### Step 4: Calculate the Eigenvalues for Each \( \lambda \)

1. **For \( \lambda_1 = 1 \):**
   \[
   \mu_1 = 3(1)^3 + 5(1)^2 - 6(1) + 2 = 3 + 5 - 6 + 2 = 4
   \]

2. **For \( \lambda_2 = 3 \):**
   \[
   \mu_2 = 3(3)^3 + 5(3)^2 - 6(3) + 2 = 3(27) + 5(9) - 6(3) + 2 = 81 + 45 - 18 + 2 = 110
   \]

3. **For \( \lambda_3 = -2 \):**
   \[
   \mu_3 = 3(-2)^3 + 5(-2)^2 - 6(-2) + 2 = 3(-8) + 5(4) - 6(-2) + 2 = -24 + 20 + 12 + 2 = 10
   \]

### Final Answer
The eigenvalues of the matrix expression \( 3A^3 + 5A^2 - 6A + 2I \) are:
\[
\boxed{4, 110, 10}
\]

Would you like further details or explanations on any of the steps? Here are some related questions:

1. How do you find the characteristic polynomial of a matrix?
2. What is the significance of triangular matrices in linear algebra?
3. Can eigenvalues be complex for certain types of matrices?
4. How do you calculate eigenvectors corresponding to these eigenvalues?
5. What are the applications of eigenvalues in different fields?

**Tip**: For triangular matrices, eigenvalues are always on the diagonal, which makes calculations much easier.

Discover how to find the eigenvalues of the matrix expression 3A³ + 5A² - 6A + 2I for the given matrix A = [(1 2 -3)( 0 3 2 )(0 0 -2)]. Step-by-step solution provided includes identifying matrix A, calculating its eigenvalues, and determining eigenvalues for the matrix expression. Suitable for advanced college-level students studying matrix algebra and eigenvalue calculations.

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