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!