0 M = NH 0 (a) Verify that the characteristic polynomial of M is X(A) = 13 – 12 -51 -3 = (1+1)2(1-3). (b) Use the Cayley-Hamilton Theorem to deduce…

Consider the following real matrix:

• Attachment 1
• Attachment 2

0M =NH0(a) Verify that the characteristic polynomial of M isX(A) = 13 – 12 -51 -3 = (1+1)2(1-3).(b) Use the Cayley-Hamilton Theorem to deduce that M is invertible andM-1 = (M2 – M – 51) .(c) Find the eigenvalues of M and the corresponding eigenspaces.(d) Find an invertible matrix P and a diagonal matrix D such that M = PDP-1.(e) Consider three differentiable functionsx = r(t) , y = y(t), z = =(t)of a real variable t linked by the following system of differential equations:+2zy++ZFind the particular solution to this system given that x(0) = 1, y(0) = 2 and=(0) = 3.