1. State the problem: Solve the system of linear equations using Gauss's Method: $$\begin{cases} 2x + 3y - z = 5 \\ 4x + y + 2z = 6 \\ -2x + 5y + 3z = 4 \end{cases}$$ 2. Use the augmented matrix and apply row operations to reduce it to row echelon form. 3. Back-substitute to find the values of $x$, $y$, and $z$. 1. State the problem: Determine the number of solutions for the system: $$\begin{cases} x + 2y - z = 3 \\ 2x + 4y - 2z = 6 \\ 3x + 6y - 3z = 9 \end{cases}$$ 2. Use Gauss's Method to reduce the system and analyze the rank of the coefficient matrix and augmented matrix. 3. Characterize the solution set (unique, infinite, or none). 1. State the problem: Find the eigenvalues of the matrix: $$A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$$ 2. Use the characteristic polynomial formula: $$\det(A - \lambda I) = 0$$ 3. Calculate the determinant and solve the quadratic equation for $\lambda$. 1. State the problem: Find the eigenvectors corresponding to the eigenvalue $\lambda = 5$ for the matrix: $$B = \begin{bmatrix} 2 & 1 \\ 1 & 4 \end{bmatrix}$$ 2. Solve the system $(B - 5I)\mathbf{v} = \mathbf{0}$ for eigenvector $\mathbf{v}$. 1. State the problem: Use Gauss's Method to solve the system: $$\begin{cases} x + y + z = 6 \\ 2x + 3y + 7z = 20 \\ -x + 4y + 5z = 10 \end{cases}$$ 1. State the problem: Determine if the matrix $$C = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$$ has eigenvalue $\lambda = 3$. 1. State the problem: Find all eigenvalues of the matrix: $$D = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$ 1. State the problem: Characterize the solutions of the system: $$\begin{cases} x - y + 2z = 1 \\ 3x - 3y + 6z = 3 \\ 2x - 2y + 4z = 2 \end{cases}$$ 1. State the problem: Find the eigenvectors of the matrix: $$E = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 4 \\ 0 & 4 & 9 \end{bmatrix}$$ 1. State the problem: Use Gauss's Method to solve: $$\begin{cases} 3x + 2y - z = 1 \\ 2x - 2y + 4z = -2 \\ -x + \frac{1}{2}y - z = 0 \end{cases}$$