The characteristic equation - Eigenvalue and Eigenvectors

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 $\det (A) = ad-bc$.

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