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The problem asks to compute \( A^5 w \), where \( A \) is a \( 3 \times 3 \) matrix with eigenvalues 2 and 1. The eigenvectors corresponding to these eigenvalues are:

- \( u = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \), corresponding to the eigenvalue \( 2 \),
- \( v = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} \), corresponding to the eigenvalue \( 1 \).

Additionally, we are given the vector \( w = \begin{bmatrix} 7 \\ 2 \\ -3 \end{bmatrix} \).

### Approach:

1. **Eigenvalue Decomposition**: Since \( u \) and \( v \) are eigenvectors of \( A \), we can express \( w \) as a linear combination of the eigenvectors:
   \[
   w = c_1 u + c_2 v
   \]
   We need to solve for \( c_1 \) and \( c_2 \).

2. **Apply the Matrix Power**: Once we find \( c_1 \) and \( c_2 \), we use the property of powers of matrices with eigenvalues:
   \[
   A^n u = \lambda_1^n u \quad \text{and} \quad A^n v = \lambda_2^n v
   \]
   So,
   \[
   A^5 w = c_1 A^5 u + c_2 A^5 v = c_1 \cdot 2^5 u + c_2 \cdot 1^5 v
   \]

Let me first solve for the constants \( c_1 \) and \( c_2 \).It seems that the approach I used doesn't directly work because we only have two eigenvectors \( u \) and \( v \), but we are dealing with a 3-dimensional vector space. Thus, the system matrix must be expanded to account for a third eigenvector or generalized eigenvector to complete the basis for the 3D space.

Since we are given two eigenvalues (2 and 1) and their corresponding eigenvectors, we need to explore a potential third eigenvector. If \( A \) is a diagonalizable matrix, the third vector must be linearly independent from \( u \) and \( v \), and it should correspond to either of the eigenvalues.

However, without explicit information on the third eigenvector, we can project \( w \) onto the known eigenvectors to approximate the solution. I'll solve it using this decomposition approach. Let me proceed with that calculation.To handle this, we'll project \( w \) onto the known eigenvectors \( u \) and \( v \). Since these eigenvectors form part of the basis, we assume that \( w \) can be expressed as a linear combination of the two eigenvectors, as follows:

\[
w = c_1 u + c_2 v
\]

I'll now solve for the coefficients \( c_1 \) and \( c_2 \) using a least-squares approximation, which will allow us to approximate \( w \) as a combination of these two vectors.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can help guide you through the manual steps of solving for \( c_1 \) and \( c_2 \) using projections, or you could retry uploading or running the calculation. Let me know how you'd like to proceed!

Let A be a 3x3 matrix. Suppose that A has eigenvalues 2 and 1, and suppose that u and v are eigenvectors corresponding to 2 and 1, respectively, where u = [1, 0, -1] and v = [2, 1, 0]. Then compute A^5 w, where w = [7, 2, -3].

This solution involves computing A^5 w for a 3x3 matrix A with given eigenvalues and eigenvectors. The problem covers concepts such as eigenvalue decomposition, matrix powers, and expressing vectors as linear combinations of eigenvectors.

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