1. (a) Find the eigenvalues and eigenvectors of A = (b) Find a matrix T (with integer entries) for which A = TJT-, for J = and where A is an eigenvalue of A. (c) With the aid of the general solution to y = Jy from the lecture notes combined with Part (b) above, find the general solution to dy1 dt = lly1 + 92 dy2 dt = -yl + 992. Use Mathematica to plot/sketch a phase portrait showing the vector field around the origin. Would you classify the origin as stable or unstable? This type of point is called an improper node. (d) The Jordan normal form of a matrix tells us that any non-diagonalisable 2 x 2-matrix A can be written in the form TJT-1. Prove that for any such matrix A that the general solution to y = Ay is y ( t ) = a(tell + elin ) + Belle where & is an eigenvector of A, o and S are constants, and where n is a generalised eigenvector, meaning (A - X1)n = 5