The characteristic equation  Eigenvalue and Eigenvectors
The characteristic equation
Lessons
Notes:
We say that a scalar $\lambda$ is an eigenvalue of an $n \times n$ matrix $A$ if and only if $\lambda$ satisfies the following characteristic equation:
$\det(A\lambda I)=0$
We say that $\det(A\lambda I)$ is a characteristic polynomial.
Useful ways to find eigenvalues
When dealing with a $2 \times 2$ matrix, use the formula $\det (A) = adbc$.
When dealing with a $3 \times 3$ matrix, use the shortcut method.
When dealing with a triangular matrix, know that the determinant is just the product of the diagonal entries.
Note that:
1. An eigenvalue is a distinct root if it has a multiplicity of 1
2. An eigenvalue is a repeated root if it has a multiplicity greater than 1

Intro Lesson
The Characteristic Equation Overview: