To find Eigen values use, | A – λI | = 0
For characteristics roots, ( A – λI )X = 0
When represents characteristics roots in matrix they called Eigen vectors.

If A is an n × n matrix, then the sum of the n eigenvalues of A is the trace of A and the product of the n eigenvalues is the determinant of A.

Eigenvalues of a Matrix and its Transpose are same.