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When it comes to analyzing the stability of an equilibrium point, eigenvalues play a crucial role. In the context of the matrix \(A\) provided, we seek to understand how a system behaves near this point. To do this, we calculate the eigenvalues. These special numbers provide insight into whether perturbations around the equilibrium will grow or die out.

To check for stability, we calculate the eigenvalues of the matrix \(A\) by finding the roots of its characteristic equation. If the real parts of the eigenvalues are negative, perturbations will decay, indicating stability. Conversely, if they are positive, perturbations grow, leading to instability. In our case, both eigenvalues are real and positive, which confirms that the equilibrium is unstable.

Classifying an equilibrium helps us describe the type of behavior that occurs around it. Our focus is on identifying whether the equilibrium point is a stable node, unstable node, saddle point, or something else.

Equilibrium classification depends mainly on the nature of the eigenvalues we calculated. Here, both eigenvalues are real and positive. This leads us to classify the equilibrium point as an **unstable node**. This indicates that any small deviation from this point in the system will likely grow over time, moving the system further away from equilibrium. Understanding whether an equilibrium is stable or unstable is essential in many fields, such as engineering and physics, to predict system behavior effectively.

The characteristic equation is central to finding the eigenvalues of a matrix. It is derived from the matrix \(A\) by subtracting \(\lambda\) times the identity matrix from \(A\) and setting the determinant equal to zero: \(\det(A - \lambda I) = 0\).

This equation gives us a polynomial that, once solved, yields the eigenvalues. For our matrix, this results in the equation \(\lambda^2 - 5\lambda + 2 = 0\). Solving this, we rely on the quadratic formula: \(\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -5\), \(c = 2\).

Finding the roots, \( \lambda_1 \) and \( \lambda_2 \), is a critical step as it allows us to characterize the equilibrium and contribute to our discussions on stability analysis and classification. Understanding the characteristic equation and its solutions busies us with the necessary insight to take further analytical steps in differential equations and systems dynamics.