Which of the following is the correct formula for finding the eigenvalues of a matrix?

To find the eigenvalues of a matrix, the correct approach is to solve the characteristic equation given by the determinant of the matrix (A - \lambda I), where (A) is the matrix in question, (I) is the identity matrix of the same size as (A), and (\lambda) is a scalar representing the eigenvalue. The formulation det(A - λI) = 0 is derived from the requirement that the matrix (A - \lambda I) must be singular for (\lambda) to be an eigenvalue, meaning that there are non-trivial solutions to the equation (Av = \lambda v).

In more detail, when calculating eigenvalues, (\lambda) is the value that, when substituted into the equation, results in the determinant being zero. This is essential because a zero determinant indicates that the corresponding linear system of equations has non-trivial solutions, hence confirming that (\lambda) is indeed an eigenvalue of (A).

The other choices do not reflect the correct formulation for finding eigenvalues:

• The first choice involves (A + \lambda I), which does not pertain to the standard eigenvalue determination process.
• The third