A's eigenvalues are 1 and 2 (real, distinct), making it non-singular and potentially diagonalizable.
Eigenvalues of a Square Matrix Satisfying A^2 - 3A + 2I = 0
The given equation, A^2 - 3A + 2I = 0, represents the characteristic polynomial of the square matrix A. This polynomial determines the eigenvalues of A, which are the special values (λ) for which the equation Ax - λx = 0 has non-trivial solutions (x ≠ 0).
Here's how we can analyze the eigenvalues of A:
Factoring the characteristic polynomial: We can factor the given polynomial as (A - 1)(A - 2) = 0. This indicates that the eigenvalues of A are 1 and 2.
Properties of eigenvalues:
Real and distinct: Since the eigenvalues are 1 and 2, both real and distinct, A will have two corresponding eigenvectors that are linearly independent.
Trace and determinant: The sum of the eigenvalues (trace) is 1 + 2 = 3, and their product (determinant) is 1 * 2 = 2.
Geometric interpretation: Eigenvalues and eigenvectors can be visualized geometrically. In this case, the eigenvectors will define directions in the n-dimensional space where A operates, and the eigenvalues represent the scaling factors applied along those directions.
Implications for A:
Non-singular: Since both eigenvalues are non-zero, A is invertible and has a full rank.
Diagonalizability: If A has n linearly independent eigenvectors, it can be diagonalized by a similarity transformation, where the diagonal elements are the eigenvalues.
Therefore, the eigenvalues of A are 1 and 2, which are both real and distinct. This implies that A is non-singular and can be diagonalized under certain conditions.