Problem 4 (10 points) Consider the following matrix A= 0 1 -9-9-5 a. Find the characteristic polynomial p(A) of the matrix A by hand. This will result in a scalar valued polynomial p(A) = aoA + a₂d² + a₂d²+ a3d³ b. Compute p(A) in matrix arithmetic by plugging the above matrix A into the polynomial p(A) = ao Aº + a₁A¹ + a₂A² + a3A³. (Hint: Recall that A°= I, the identity matrix, similar to how X0 = 1.) c. Compute the eigenvalues of A. (You can do this with the Matlab and Octave roots() function.) d. For each of the eigenvalues A₁, A2, A3, compute the associated eigenvectors using the following template code in matlab a "m" script file. A = %fill in matrix A here. I = eye (size (A)) s = %fill in eigenvalue here. rref (A - s*I) in Matlab or Octave. This gives you the RREF for the linear system Be=0 where BA-sI. The solution vectors v are eigenvectors. It is ok to leave the default number of decimal places for computation. Provide the printed program output for your homework as well as the corresponding eigenvector for your homework. e. Using the computed eigenvectors of part d. compute the diagonal matrix D= [v₁ | 12 | 13] ¹ A[v₁ | 1₂ | 13] in Octave or matlab. Provide the program output of D for your homework. This can be done with the template code in matlab a ".m" script file. A %fill in matrix A here. vi = %fill in eigenvector here v2 = %fill in eigenvector here v3 %fill in eigenvector here inv ( [v1, v2,v3]) *A* [v1, v2,v3] D=