Hello, please help me with these questions, and give explanations for the answers. Please don't use cursive as I can't read that, and please don't type the explanations. These are practice examples I saw on the internet. No reference, unfortunately.

1 3. Compute the determinant of the Vandermonde matrix 1 b b2 C 4. The characteristic polynomial of a matrix A is XA (A) = (X -1) (X -2)-(X + 2). (a) What is the order (size) of A? (b) Find the eigenvalues of A. (c) Find the trace of A. (d) Find the determinant of A.6. Consider A = 1 1 1 1 (a) Show that r = - i is an eigenvector of A and find its comresponding eigenvalue. (b) Show that 0 is an eigenvalue of A. (c) Using the trace of A, verify that 0 is an eigenvalue of A of multiplicity 2. (d) Find two linearly independent eigenvectors of A corresponding to eigenvalue 0.5. (7 points) Let A (a) (4 pts) Orthogonally diagonalize A, i.e., find an orthogonal matrix Q and a diagonal matrix A such that A = QAQT. (b) (3 pts) Use part (a) to find A*, where k is a positive integer. (Your answer should be a single matrix)