# The characteristic equation

##### Intros
###### Lessons
1. The Characteristic Equation Overview:
2. What is the Characteristic Equation?
$\det (A-\lambda I)=0$
• The characteristic polynomial is $\det (A-\lambda I)$
3. Finding the Eigenvalue
• The eigenvalues of a $2 \times 2$ matrix
• The eigenvalues of a $3 \times 3$ matrix
• The eigenvalues of a triangular matrix
4. Shortcut to Determinants of Matrices
$2 \times 2$ matrices
$3 \times 3$ matrices
• Triangular matrices
5. Eigenvalue with Multiplicity
• Distinct eigenvalues
• Repeated eigenvalues
##### Examples
###### Lessons
1. Finding the Characteristic Polynomial
Find the characteristic polynomial of $A$ if:
1. Find the characteristic polynomial of $A$ if:
1. Finding the Eigenvalues of a $2 \times 2$ matrix
Find all the eigenvalues of the matrix
State their multiplicities, and what type of eigenvalues they are.
1. Finding the Eigenvalues of a $3 \times 3$ matrix
Find all the eigenvalues of the matrix
State their multiplicities, and what type of eigenvalues they are.
1. Finding the Eigenvalues of a triangular matrix
Find all the eigenvalues of the matrix
State their multiplicities, and what type of eigenvalues they are.
1. Proofs dealing with the Characteristic Equation
Show that $A$ and $A^T$ has the same characteristic polynomials.
1. Suppose $A$ is an $n \times n$ triangular matrix where all the diagonal entries are $c$. Then the characteristic polynomial is
$p(\lambda)=(c-\lambda)^n$