the eigen vectors of a 3x3 matrix A corresponding to the eigenvalues 1,-1,2 are
[1 1 0]  , [ 1 0 1] , [3 1 1 ] (written in column matrix)  respectively . Find the matrix A and hence find A^5

To find the matrix \( A \), we need to use the given eigenvalues and their corresponding eigenvectors.

### Step 1: Write the matrix of eigenvectors \( P \)
Given the eigenvectors \( v_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \), \( v_2 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \), and \( v_3 = \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix} \), we can write the matrix \( P \) whose columns are these eigenvectors:
\[
P = \begin{bmatrix}
1 & 1 & 3 \\
1 & 0 & 1 \\
0 & 1 & 1
\end{bmatrix}
\]

### Step 2: Write the diagonal matrix \( D \)
The eigenvalues corresponding to these eigenvectors are \( 1 \), \( -1 \), and \( 2 \), so the diagonal matrix \( D \) is:
\[
D = \begin{bmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 2
\end{bmatrix}
\]

### Step 3: Use the formula \( A = P D P^{-1} \)
The matrix \( A \) can be found using the formula:
\[
A = P D P^{-1}
\]
We already have \( P \) and \( D \), so we just need to compute \( P^{-1} \), and then find \( A \).

#### Find \( P^{-1} \)
To compute \( P^{-1} \), we calculate the inverse of \( P \).

Let me compute it.

The matrix \( A \) is:
\[
A = \begin{bmatrix}
6 & -5 & -7 \\
1 & 0 & -1 \\
3 & -3 & -4
\end{bmatrix}
\]

### Step 4: Compute \( A^5 \)
To find \( A^5 \), we can use the fact that:
\[
A^5 = P D^5 P^{-1}
\]
Where \( D^5 \) is the diagonal matrix with the fifth powers of the eigenvalues on the diagonal. That is:
\[
D^5 = \begin{bmatrix}
1^5 & 0 & 0 \\
0 & (-1)^5 & 0 \\
0 & 0 & 2^5
\end{bmatrix}
= \begin{bmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 32
\end{bmatrix}
\]

Now, let's calculate \( A^5 \).

The matrix \( A^5 \) is:
\[
A^5 = \begin{bmatrix}
96 & -95 & -97 \\
31 & -30 & -31 \\
33 & -33 & -34
\end{bmatrix}
\]

Would you like more details on any part of the process? Here are some related questions:

1. How do you find the inverse of a matrix?
2. What are the properties of eigenvalues and eigenvectors?
3. How does diagonalization simplify computing powers of a matrix?
4. What is the geometric interpretation of eigenvalues and eigenvectors?
5. How can eigenvalues help in determining the stability of a system?

**Tip:** Diagonalizing a matrix simplifies the computation of higher powers and exponentials of matrices, which is particularly useful in linear algebra and differential equations.

The eigenvectors of a 3x3 matrix A corresponding to the eigenvalues 1, -1, 2 are [1 1 0], [1 0 1], [3 1 1] (written in column matrix) respectively. Find the matrix A and hence find A^5.

This solution explains how to find a matrix A given its eigenvectors and corresponding eigenvalues, and how to compute A^5 using diagonalization. The eigenvalues are 1, -1, and 2 with eigenvectors [1, 1, 0], [1, 0, 1], and [3, 1, 1]. A step-by-step solution is provided for undergraduate linear algebra students.

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