To find the eigenvalues and eigenvectors of matrix $A$, we follow these steps:

Matrix $A$:

$A = \begin{bmatrix} -2 & 2 & -3 \\ 2 & 1 & 6 \\ -1 & -2 & 0 \end{bmatrix}$

Steps:

1. Find the Eigenvalues:

   • To find the eigenvalues $\lambda$ of $A$, solve the characteristic polynomial equation $\det(A - \lambda I) = 0$, where $I$ is the identity matrix of the same size as $A$.

2. Solve for Eigenvectors:

   • For each eigenvalue $\lambda$, solve the system $(A - \lambda I) \vec{v} = 0$ to find the corresponding eigenvector $\vec{v}$.

Let me calculate these values.The eigenvalues and eigenvectors of the matrix $A$ are as follows:

Eigenvalues:

1. $\lambda_1 = -3$
2. $\lambda_2 = 1 + 2.8284i$ (complex eigenvalue)
3. $\lambda_3 = 1 - 2.8284i$ (complex conjugate of $\lambda_2$)

Eigenvectors:

1. For $\lambda_1 = -3$: $\vec{v}_1 = \begin{bmatrix} -0.8944 \\ 0.4472 \\ 0 \end{bmatrix}$

2. For $\lambda_2 = 1 + 2.8284i$: $\vec{v}_2 = \begin{bmatrix} 0 + 0.5i \\ -0.7071 \\ 0 - 0.5i \end{bmatrix}$

3. For $\lambda_3 = 1 - 2.8284i$: $\vec{v}_3 = \begin{bmatrix} 0 - 0.5i \\ -0.7071 \\ 0 + 0.5i \end{bmatrix}$

The presence of complex eigenvalues indicates that this matrix has no real eigenvectors associated with the complex eigenvalues.

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Related Questions:

1. How do we interpret complex eigenvalues and eigenvectors geometrically?
2. What role do eigenvalues play in matrix diagonalization?
3. How does the determinant of $A$ relate to its eigenvalues?
4. Can we find the eigenvalues of a matrix without computing the characteristic polynomial?
5. How would we compute the eigenvalues and eigenvectors of larger matrices?

Tip:

When solving for eigenvalues, complex eigenvalues often appear in conjugate pairs for real matrices.