To find the eigenvalues of a 2x2 matrix, you can use the characteristic polynomial method.

  Let's say you have a 2x2 matrix A:

  A = [a b]

   [c d]

  Step 1: Calculate the determinant of the matrix A.

  det(A) = ad - bc

  Step 2: Set up the characteristic equation by subtracting λ (lambda) from the diagonal elements.

  det(A - λI) = 0

  Here, λ is the eigenvalue and I is the identity matrix.

  For the given matrix A, the characteristic equation will be:

  det(A - λI) = (a - λ)(d - λ) - bc = 0

  Step 3: Solve the equation for λ in order to find the eigenvalues.

  Expand the equation:

  ad - aλ - dλ + λ^2 - bc = 0

  Rearrange the terms:

  λ^2 - (a + d)λ + ad - bc = 0

  This is a quadratic equation in terms of λ. Use the quadratic formula to solve for λ:

  λ = (-b ± √(b^2 - 4ac))/2a

  Where a = 1, b = -(a + d), and c = ad - bc.

  By substituting these values, you can find the two eigenvalues of the matrix A.