There isn't really any general way to find a matrix's eigenvalues, other than explicitly calculating (that is, assuming A is our matrix) and factoring. The roots of are the eigenvalues of A.

Also, I don't really know what you're referring to by the Rule of Sarrus.

Why don't you post some of your attempts and we will help you.

P.S.

There are some theorems that help finding eigenvalues, such as:

(*) If the sum of elements of each row in the matrix A is equal to n for some constant n, then n is an eigenvalue with corresponding eigenvector

Similarly, if the sum of elements of eachcolumnis equal to k for some constant k, then k is also an eigenvalue, however we do not know if will be an eigenvector.

(*) If A is of order and , then 0 is an eigenvalue of A.

Similarly, if then is an eigenvalue of A, with geometric multiplicity