Prоblem 1. (20 pts) Let (A=begin{bmаtrix}-3& 4\-2&3end{bmаtrix}). (а) (8 pts) Find (A^k) fоr generic (k). (b) (8 pts) Find (e^{At}) fоr (tge 0). (c) (4 pts) Find (alpha) and (beta) so that (cos(A)=alpha A + beta I) where (I) is the identity matrix. Problem 2. (20 pts) Let A∈ℝ4x4{"version":"1.1","math":"A∈ℝ4x4"} be given by [v1v2v3v4]⏟T∈ℝ4x4 0-110-1-1⏟S{"version":"1.1","math":"[v1v2v3v4]⏟T∈ℝ4x4 0-110-1-1⏟S"}w1Tw2Tw3Tw4T⏟T-1{"version":"1.1","math":"w1Tw2Tw3Tw4T⏟T-1"} Note that the middle matrix (S), though block diagonal, is not in standard Jordan form. (a) (3 pts) For the continuous-time LTI system (dot x=Ax), determine its stability. (b) (3 pts) Given x(0)=v1-2v3+3v4∈ℝ4{"version":"1.1","math":"x(0)=v1-2v3+3v4∈ℝ4"}, find the solution (x(t)) of (dot x=Ax). (c) (6 pts) Find initial states (x(0)) so that the corresponding solution (x(t)) satisfies as (ttoinfty) (i) converges to 0; (ii) diverges to infinity; (iii) neither (i) nor (ii). For each case, if it is possible, find one such (x(0)neq 0); otherwise, explain why. Next consider the discrete-time LTI system (x[k+1]=Ax[k]) with the same (A) matrix given above. (d) (2 pts) Determine the stability of the discrete-time system (x[k+1]=Ax[k]). (e) (6 pts) Find initial states (x[0]) so that the corresponding solution (x[k]) satisfies as (ktoinfty) (i) converges to 0; (ii) diverges to infinity; (iii) neither (i) nor (ii). For each case, if it is possible, find one such (x[0]neq 0); otherwise, explain why. Problem 3. (20 pts) Consider an LTI system (dot x=Ax) with the matrix (Ainmathbb R^{2times 2}) unknown. Suppose two experiments are carried out. In Experiment 1, for the initial state x(0)=11{"version":"1.1","math":"x(0)=11"} it is measured that x(1)=22{"version":"1.1","math":"x(1)=22"}. In Experiment 2, for the initial state x(0)=12{"version":"1.1","math":"x(0)=12"} we have x(1)=34{"version":"1.1","math":"x(1)=34"}. (a) (12 pts) Based on the two experiments, determine (e^A) and (A). (b) (8 pts) Suppose x(0)=01{"version":"1.1","math":"x(0)=01"}. For the resulting solution (x(t)), find (x(2)) and (x(0.5)), respectively. Problem 4. (20 pts) Consider a system (dot x=Ax) with x∈ℝ3{"version":"1.1","math":"x∈ℝ3"} and an unknown matrix A∈ℝ3x3{"version":"1.1","math":"A∈ℝ3x3"}. Suppose under the initial condition x(0)={"version":"1.1","math":"x(0)="}001{"version":"1.1","math":"001"}, the system has the solutionx(t)={"version":"1.1","math":"x(t)="}e-t-1e-7t-e-t2-e-7t{"version":"1.1","math":"e-t-1e-7t-e-t2-e-7t"}. (a) (6 pts) Identify all of the eigenvalues of matrix (A) and rewrite the above given (x(t)) as a linear combination of the modes associated with these eigenvalues. (b) (4 pts) What are the eigenvectors corresponding to these eigenvalues? (c) (5 pts) Determine the matrix (A). You do not need to simplify the answer to a single matrix. (d) (5 pts) To have (lim_{ttoinfty} x(t)=0), what exact conditions should a generic (x(0)) satisfy? Problem 5. (20 pts) Consider the following LTV system x˙(t)=A(t)x(t)={"version":"1.1","math":"x˙(t)=A(t)x(t)="}-2t0t2{"version":"1.1","math":"-2t0t2"}x(t), {"version":"1.1","math":"x(t), "} x(0)={"version":"1.1","math":"x(0)="}11{"version":"1.1","math":"11"}. Find (x(t)) for all (tge 0). Note that (A(s)A(t)neq A(t)A(s) ) for general (s) and (t). Congratulations, you are almost done with Midterm Exam 1. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope to submit your work: Midterm Exam 1 Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.