Finding Eigenvalues of a 2x2 Matrix

Consider the matrix $A$, with entries $-3, -4; 5, 6$. We must first find the eigenvalues, which means we must solve for the values of $\lambda$ that satisfy this expression, where the determinant of $A - \lambda I = 0$.

In this problem, we are tasked with finding the eigenvalues of a matrix. This process is a fundamental operation in linear algebra and has applications in various fields such as physics, computer science, and engineering. The eigenvalues provide insights into the properties of the transformation that the matrix represents, particularly the scaling factors in the direction of its eigenvectors. The problem specifically involves solving a characteristic equation that is formed by subtracting a scalar lambda, multiplied by the identity matrix, from the matrix A, and setting the determinant of this result to zero. Solving this equation helps identify the values of lambda that allow existence of non-zero vectors satisfying the transformation conditions.

Understanding how to manipulate determinants and implement the characteristic equation to find eigenvalues is crucial for grasping more complex topics such as diagonalization and matrix exponentiation. The ability to determine eigenvalues also aids in assessing stability and dynamic behavior in systems of differential equations and in principal component analysis for data dimensionality reduction. As students delve into this problem, they should remember that identifying eigenvalues is not just about computation but about understanding underlying geometric transformations represented by matrices. This conceptual foundation is important for tackling more advanced problems in linear algebra.