Form the characteristic matrix (A – lambda I)
Compute the determinant (det(A – lambda I))
Set the characteristic polynomial equal to zero
Solve (det(A – lambda I) = 0) for (lambda)
The solutions (lambda) are the eigenvalues
For a (2 times 2) matrix (begin{pmatrix} a & b \ c & d end{pmatrix}), solve ((a-lambda)(d-lambda) – bc = 0)
For a (3 times 3) matrix, compute the determinant of (A – lambda I) and solve the resulting cubic equation
For larger matrices, use algebraic methods, numerical methods, or software to solve the characteristic equation