For a matric 2,-2,3,1,1,1, 1,3 - 1 the eigonvalue is - 2 find the eigonvector 

Given the matrix \( A \) and an eigenvalue \( \lambda = -2 \), the eigenvector \( \mathbf{v} \) corresponding to this eigenvalue is a non-zero vector that satisfies the equation:

\[
(A - \lambda I)\mathbf{v} = 0
\]

Where \( I \) is the identity matrix of the same dimension as \( A \).

Let's proceed with the calculation.

Given matrix \( A \) as:

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

The eigenvalue \( \lambda = -2 \).

We need to find the matrix \( A - \lambda I \):

\[
A - (-2)I = A + 2I = \begin{pmatrix} 
2 & -2 & 3 \\
1 & 1 & 1 \\
1 & 3 & -1 
\end{pmatrix} + 2 \begin{pmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{pmatrix} = \begin{pmatrix} 
4 & -2 & 3 \\
1 & 3 & 1 \\
1 & 3 & 1 
\end{pmatrix}
\]

Now, solve the equation:

\[
\begin{pmatrix} 
4 & -2 & 3 \\
1 & 3 & 1 \\
1 & 3 & 1 
\end{pmatrix} \begin{pmatrix} 
x_1 \\
x_2 \\
x_3 
\end{pmatrix} = \begin{pmatrix} 
0 \\
0 \\
0 
\end{pmatrix}
\]

This represents a system of linear equations:

\[
4x_1 - 2x_2 + 3x_3 = 0
\]
\[
x_1 + 3x_2 + x_3 = 0
\]
\[
x_1 + 3x_2 + x_3 = 0
\]

Notice that the last two equations are identical, which simplifies the system to:

\[
4x_1 - 2x_2 + 3x_3 = 0
\]
\[
x_1 + 3x_2 + x_3 = 0
\]

We can solve this system by expressing \( x_1 \) and \( x_3 \) in terms of \( x_2 \). Let's solve it:

From the second equation:

\[
x_1 = -3x_2 - x_3
\]

Substitute into the first equation:

\[
4(-3x_2 - x_3) - 2x_2 + 3x_3 = 0
\]

Simplify:

\[
-12x_2 - 4x_3 - 2x_2 + 3x_3 = 0 \Rightarrow -14x_2 - x_3 = 0
\]

Thus:

\[
x_3 = -14x_2
\]

Substituting \( x_3 = -14x_2 \) back into \( x_1 = -3x_2 - x_3 \):

\[
x_1 = -3x_2 - (-14x_2) = 11x_2
\]

So the eigenvector \( \mathbf{v} \) is:

\[
\mathbf{v} = x_2 \begin{pmatrix} 
11 \\
1 \\
-14 
\end{pmatrix}
\]

This is the eigenvector corresponding to the eigenvalue \( \lambda = -2 \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you find the characteristic equation of a matrix?
2. What is the significance of an eigenvalue in a real-world application?
3. Can a matrix have more than one eigenvector for a single eigenvalue?
4. What happens if an eigenvalue is repeated?
5. How do you interpret the geometric meaning of eigenvectors and eigenvalues?

**Tip:** Always verify your solutions to eigenvalue problems by substituting back into the original matrix equation.

Learn how to find the eigenvector corresponding to eigenvalue -2 in matrix A. Detailed step-by-step solution including matrix operations and eigenvalue equation. Suitable for advanced undergraduate students studying linear algebra.

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