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:

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.