(a) Find the eigenvalues of the matrix A _ 24 and find an eigenvector corresponding to each eigenvalue. Hence find an invertible matrix P and a diagonal matrix D such that P-1AP = D. (b) Suppose that a system of differential equations can be written in the form f' (t) = Af (t ) + q where A is diagonalisable and q is a constant vector. Show that we can define a vector u(t) which allows us to rewrite this system of differential equations as u'(t) = Du(t) + P-1q for some matrices P and D. (c) Using parts (a) and (b), find the functions f(t) and g(t) that satisfy the system o differential equations f' (t) = 11f(t) - 4g(t) +4 g' (t) = 24f(t) - 9g(t) + 12 and the initial conditions f (0) = 7 and g(0) = 19