The question involves analyzing the direction field of a linear system and determining the nature of the system's equilibrium points based on eigenvalues.

Observing the Direction Field:

The field shows a saddle point at the origin (0,0). This is because the arrows indicate movement away from the origin in some directions and towards the origin in others, a classic feature of a saddle point.

Eigenvalue Analysis:

A saddle point indicates that the system has two real eigenvalues with opposite signs. One eigenvalue will be positive (indicating unstable behavior in one direction), and the other will be negative (indicating stable behavior in another direction).

Conclusion:

The correct answer is:

• The eigenvalues have opposite sign.

Would you like more details on how eigenvalues relate to the direction field, or any other aspect of the problem?

Here are five related questions:

1. How can you determine stability from eigenvalues?
2. What is the difference between a node and a saddle point in a direction field?
3. How do complex eigenvalues affect the appearance of direction fields?
4. How would the direction field change if both eigenvalues were positive?
5. What is the significance of the origin in linear systems like this?

Tip: For linear systems, analyzing the trace and determinant of the matrix can quickly indicate the type of equilibrium at the origin.