need help 9-12

9. a. If A is a diagonal matrix, and all of the eigenvalues of A are equal to 2 or - 2, explain why A? = 4I. b. If A is a diagonalizable matrix and all of the eigenvalues of A are equal to 2 or - 2, explain why A2 - 41. 10. If A and Bare square matrices of the same size, one says A and B are similar if there is on invertible matrix P such that A= PBP ? (so, for example, A is didgenalizable if A is similar to a diagonal matrix.) If A and B are similar matrices and I is an eigenvalue of B, explain why I is also an eigenvalue of A. 14. If A and B are similar matrices, and I is an eigenvector of By, is & algo on eigenvector of A? Explain why or why not. 12. a. Find a nonzero solution to the system of ODE, x () = x (+) ty (t ) y' ( ) = 4 (t ) . Do the methods we have learned in class allow us to find a basis for the vector space of solutions of this system? 12. b , Show that is a solution to the system P - in part a . C . Using part b, Find a basis for the vector space of solutions to the system of ODE in Fort a