91Ó°ÊÓ

To begin understanding eigenvalues, it's essential to grasp what a characteristic polynomial is. In essence, this polynomial is your golden ticket to finding eigenvalues. When scrutinizing a matrix, like the one given in our exercise, unleashing its characteristic polynomial involves a bit of matrix algebra magic.

We take the original matrix and perform a slight alteration: for each element on the main diagonal, we subtract a variable often denoted as \(\text{\lambda}\), representing our eventual eigenvalues. This concoction results in a new matrix, \(A - \lambda I\), where \(I\) stands for the identity matrix, and \(A\) is our original matrix. The determinant of this modified matrix is our coveted characteristic polynomial.

Tackling this step-by-step means calculating the determinant of a two-by-two matrix, like so: \(\det(\begin{bmatrix} 5 - \lambda & -4 \ 2 & -1 - \lambda \end{bmatrix})\), which later transforms into a polynomial equation. The roots of this polynomial, deduced by setting it equal to zero, are the matrix's eigenvalues, the very core of our quest.

Pondering how to calculate the determinant? It's not as daunting as it seems. When faced with a two-by-two matrix, as seen in our exercise, the formula is straightforward: \(\det(\begin{bmatrix} a & b \ c & d \end{bmatrix}) = ad - bc\). Now, let's apply this to our specific case: \(\det(\begin{bmatrix} 5 - \lambda & -4 \ 2 & -1 - \lambda \end{bmatrix}) = (5 - \lambda)(-1 - \lambda) - (-4)(2)\).

Ensuring You're on the Right Path

As we follow through with the determinant calculation, we're led to a quadratic equation. It's essential to expand meticulously and to simplify thoroughly, at which point our equation will be ready for factorization or for the quadratic formula - whichever you prefer. Whichever the approach, the goal is to excavate the eigenvalues, those mystical numbers that signify so much in the realm of linear algebra.

Stumbling upon the term 'algebraic multiplicity' might seem like encountering a mythical beast, but fear not. It's simply a name for the number of times an eigenvalue appears as a root of the characteristic polynomial. Think of it like counting how many times a specific number shows up in our polynomial puzzle.

Eigenvalues: The Repeat Offenders

Upon solving the polynomial and finding that our eigenvalues are \(\lambda = 1\) and \(\lambda = 3\), we determine that each appears exactly once. Their algebraic multiplicity is therefore 1. In more complex scenarios, an eigenvalue could repeat, indicating a higher multiplicity. Multiplicity isn't just a headcount — it has profound implications on the geometric nature of our matrix and can signal deeper patterns at play within the linear transformation it represents.