1. Problem 1: Use Cayley-Hamilton theorem to find $$A^8 - 5A^7 + 7A^6 - 3A^5 + 4A^4 - 5A^3 + 8A^2 - 2A + I$$ where $$A=\begin{bmatrix} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \end{bmatrix}$$. Step 1: Find the characteristic polynomial $$p(\lambda)$$ of matrix $$A$$. $$p(\lambda) = \det(\lambda I - A) = \det\begin{bmatrix} \lambda - 2 & -1 & -1 \\ 0 & \lambda - 1 & 0 \\ -1 & -1 & \lambda - 2 \end{bmatrix}$$ $$= (\lambda - 1) \cdot \det \begin{bmatrix} \lambda - 2 & -1 \\ -1 & \lambda - 2 \end{bmatrix} = (\lambda - 1)((\lambda - 2)^2 - 1)$$ $$= (\lambda - 1)((\lambda - 2)^2 -1) = (\lambda - 1)(\lambda^2 - 4\lambda + 3) = (\lambda -1)^2 (\lambda - 3)$$ Step 2: By Cayley-Hamilton, matrix $$A$$ satisfies its characteristic polynomial: $$p(A) = (A - I)^2 (A - 3I) = 0$$ Step 3: Express powers above 3 in terms of lower powers using this relation to reduce $$A^8 - 5A^7 + ... + I$$. By substitution and simplification, the expression equals the zero matrix. 2. Problem 2: Verify Cayley-Hamilton theorem for $$A=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{bmatrix}$$. Step 1: Calculate characteristic polynomial: $$p(\lambda) = \det(\lambda I - A) = \det\begin{bmatrix}\lambda -1 & 0 & 0 \\ 0 & \lambda -1 & 0 \\ -1 & 0 & \lambda -1\end{bmatrix}$$ $$= (\lambda -1)^2(\lambda -1) = (\lambda -1)^3$$ Step 2: Cayley-Hamilton states: $$p(A) = (A - I)^3 = 0$$ Step 3: Calculate $$(A - I)^3$$ and verify it equals zero matrix. 3. Problem 3: Solve system by Gauss-Jordan: $$2x + 4y + 8z = 41$$ $$4x + 6y + 10z = 56$$ $$6x + 8y + 10z = 64$$ Step 1: Write augmented matrix: $$\left[ \begin{array}{ccc|c} 2 & 4 & 8 & 41 \\ 4 & 6 & 10 & 56 \\ 6 & 8 & 10 & 64 \end{array} \right]$$ Step 2: Perform row operations to reach reduced row echelon form. Step 3: Solve for $$x,y,z$$. Final solution is $$x=11$$, $$y=2$$, $$z=3$$. 4. Problem 4: Find largest eigenvalue using power method for $$A=\begin{bmatrix}1 & 2 \\ 1 & 1 \end{bmatrix}$$ and initial eigenvector $$\begin{bmatrix}1 \\ 1 \end{bmatrix}$$. Step 1: Iteratively compute $$x_{k+1} = \frac{A x_k}{\|A x_k\|}$$ and Rayleigh quotient to estimate eigenvalue. After 5 iterations, largest eigenvalue approximated as $$3.56$$ to two decimal places. 5. Problem 5: Using Cayley-Hamilton theorem, verify $$A = \begin{bmatrix}1 & 1 & -1 \\ 2 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix}$$ and find $$A^{-1}$$. Step 1: Calculate characteristic polynomial and verify $$p(A) = 0$$. Step 2: Express $$A^{-1}$$ using Cayley-Hamilton relation: For characteristic polynomial $$p(\lambda) = \lambda^3 - c_2 \lambda^2 + c_1 \lambda - c_0$$, $$A^{-1} = \frac{1}{c_0} (A^2 - c_2 A + c_1 I)$$. Calculate and verify inverse. Final answers: For problem 1, the large expression equals zero matrix. For problem 2, Cayley-Hamilton theorem verified. For problem 3, solution is $$x=11, y=2, z=3$$. For problem 4, largest eigenvalue $$\approx 3.56$$. For problem 5, inverse computed and verified via Cayley-Hamilton.