Form the matrix (A – lambda I), where (I) is the identity matrix and (lambda) is a scalar
Compute the determinant (det(A – lambda I))
Set the characteristic equation (det(A – lambda I) = 0)
Solve the resulting polynomial equation for (lambda)
The solutions (lambda) are the eigenvalues of the matrix
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 or larger, expand the determinant and solve the characteristic polynomial
Verify each eigenvalue by checking whether ((A – lambda I)x = 0) has a nonzero solution