Let A be a square matrix and k be a positive integer. Show that if A is an eigenvalue of A then X* is an eigenvalue of Ak.

Let A be an n x n matrix and let λ be an eigenvalue of A. The collection of all eigenvectors corresponding to A, together with the zero vector, is called the eigenspace of A and is denoted by EA. qs)Why is it important that this definition includes the phrase "together with the zero vector"? (i.e., what would go wrong if this phrase was not included?)

Let A be a nxn symmetric matrix with real entries. (a) Show that if A is an eigenvector for A, then X must be a real number. (b) If A1 and A2 are two eigenvalues for A with vị and v2 eigenvectors, respectively. Then vi and a2 are orthogonal.