[Matlab & Math] I need to derive an expression to find eigenvalues of a matrix in terms of trace() and det() function

Calculate eigen values and eigen functions of L power of 2 and Lz

Show that similar (~) matrices have the same eigen values. Similar means: A ~ B if there is a non-singular matrix P and A = P^(-1) B P. Please use typefont or at least legible handwriting, and keep it short and simple with clear explanations. Thank you!

Find the characteristic polynomial, and the eigenvalues, of the matrix A below. For each eigenvalue ^, find a basis for the eigenspace.

is an eigenvector of the matrix 2 What is its eigenvalue? Eigenvalue =

Show that the eigenvalues of a real symmetric matrix A = A^T arealways real (use the conjugate transpose).

The eigenvalue should be 2-2i, which with 8 would create a 1/4 ratio. Also, the row [1 0] is the first, not second row as I have it. I'm not sure at which step I've made a mistake. Any help would be appreciated