5. (a) Find the eigenvalues of the matrix A = and find an eigenvector corresponding to each eigenvalue. Hence find an invertible matrix P and a diagonal matrix D such that P-'AP = D. (b) Suppose that a system of difference equations can be written in the form where A is diagonalisable and q is a constant vector. Show that, we can define a vector u, which allows us to rewrite this system of difference equations as u - Dugit p q for some matrices P and D. (c) Using parts (a) and (b), find the sequences z, and y, that satisfy the system of difference equations 1 = 110 1- 4y-1+4 31 = 24.T-1 - 991-1 + 12 m pap and the initial conditions ro = 3 and yo = 7