Find the eigen values & eigen vectors of the matrix 
 ൭
4 2 −2
−5 3 2
−2 4 1
൱

To find the eigenvalues and eigenvectors of the given matrix \( A \), let's proceed with the following steps.

The matrix \( A \) is:

\[
A = \begin{pmatrix} 
4 & 2 & -2 \\
-5 & 3 & 2 \\
-2 & 4 & 1 
\end{pmatrix}
\]

### Step 1: Find the Eigenvalues

The eigenvalues \( \lambda \) are found by solving the characteristic equation:

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

where \( I \) is the identity matrix and \( \lambda \) is a scalar. So we compute the determinant of the matrix \( A - \lambda I \).

\[
A - \lambda I = \begin{pmatrix} 
4 - \lambda & 2 & -2 \\
-5 & 3 - \lambda & 2 \\
-2 & 4 & 1 - \lambda 
\end{pmatrix}
\]

Now, calculate the determinant of this matrix:

\[
\text{det}(A - \lambda I) = \text{det} \begin{pmatrix} 
4 - \lambda & 2 & -2 \\
-5 & 3 - \lambda & 2 \\
-2 & 4 & 1 - \lambda 
\end{pmatrix}
\]

I'll compute this determinant and find the eigenvalues. Let me do the math.The characteristic equation is:

\[
-\lambda^3 + 8\lambda^2 - 17\lambda + 10 = 0
\]

Now, solving this cubic equation will give us the eigenvalues. Let me compute them next.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can solve the cubic equation manually or using a calculator to find the eigenvalues of the matrix.

The cubic equation to solve is:

\[
-\lambda^3 + 8\lambda^2 - 17\lambda + 10 = 0
\]

Once you find the eigenvalues, we can proceed to calculate the eigenvectors.

Do you have any questions or would you like more details on how to proceed manually?

Here are some related questions to explore:
1. How do you solve cubic equations using factorization or numerical methods?
2. What are the properties of eigenvalues in terms of matrix transformations?
3. Can the sum and product of eigenvalues tell us something about the matrix?
4. What are the geometric interpretations of eigenvectors?
5. How does matrix symmetry affect the eigenvalues and eigenvectors?

**Tip:** Always check if the determinant of the matrix is zero. If it is, one of the eigenvalues is \( \lambda = 0 \).

Find the eigenvalues and eigenvectors of the matrix: 
[ 4 2 -2 ]
[ -5 3 2 ]
[ -2 4 1 ]

Learn how to find the eigenvalues and eigenvectors of the matrix A = [ 4 2 -2; -5 3 2; -2 4 1 ] by solving the characteristic equation and determining the roots of the cubic equation -λ^3 + 8λ^2 - 17λ + 10 = 0. Suitable for advanced high school or undergraduate students.

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