I see that you've uploaded an image containing problems related to eigenvalues and eigenvectors, specifically determining eigenvalues and the basis for eigenspaces of given matrices. Let me extract and analyze the content.

Give me a moment to process this.The image contains several tasks involving eigenvalues and eigenvectors. Here's a breakdown:

Key Content:

1. Definition of Eigenspace:

   • Vectors $x$ related to eigenvalue $\lambda$ satisfy $(A - \lambda I)x = 0$.
   • The solution space for this equation is called the eigenspace of matrix $A$.

2. Examples to Solve:

   • Determine the basis for the eigenspace of the given matrices:
     • Matrix 1: $A = \begin{bmatrix} 1 & -1 \\ 1 & 3 \end{bmatrix}$
     • Matrix 2: $B = \begin{bmatrix} 1 & 0 & 5 \\ 0 & 2 & 0 \\ 3 & 0 & 3 \end{bmatrix}$
     • Matrix 3: $D = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix}$

3. Homework (TUGAS 8):

   • Find eigenvalues and bases for eigenspaces for the following matrices:
     • Matrix 1: $\begin{bmatrix} 3 & 0 \\ 8 & 1 \end{bmatrix}$
     • Matrix 2: $\begin{bmatrix} -2 & -7 \\ 1 & 2 \end{bmatrix}$
     • Matrix 3: $\begin{bmatrix} 4 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 1 \end{bmatrix}$
     • Matrix 4: $\begin{bmatrix} 3 & 0 & -5 \\ \frac{1}{5} & -1 & 0 \\ 1 & 1 & -2 \end{bmatrix}$
     • Matrix 5: $\begin{bmatrix} 0 & 0 & 2 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Would you like me to solve one of these examples step by step, or should I provide solutions for all?