Problem 1:   

(a) Let 

Hand calculate A+B, A-B, AB, BA, CA, CB, and AC^T.

(b) Use MATLAB to verify your calculations.

Problem 2:

(a) Let Hand calculate |A|, r_A, and A^-1 if it exists.

(b) Hand calculates the eigenvalues and eigenvectors of matrix A.

(c) Use MATLAB to verify your calculations.

Problem 3: (a) Use Gaussian elimination to solve for x₁ ∼ x₃, if .

(b) Use MATLAB to verify your calculations.

Problem 4: Consider x₁ = [1  2  3]^T, x₂ = [1  -2  3]^T, x₃ = [0  1  1]^T.

(a) Show that this set is linearly independent.

(b) Outline the steps on how you can use MATLAB to verify your calculations.

(c) Use MATLAB to verify your calculations.

Problem 5: Scalar case 

Verify that x(t) = e^{t_0∫^ta(τ)dτ} x(t₀) + _{t_0}∫^te^{τ_∫^ta(ζ)dζ} b(τ)u(τ)dτ is the solution of x^· = a(t)x(t)+b(t)u(t).

Problem 6: Matrix case 

Verify that x(t) = e^(t-t­_0)Ax(t₀) + _{t_0}∫^te^(t-τ)AB(τ)u(τ)dτ is the solution of x^· = Ax + B(t)u(t).

Problem 7: Prove that the system presented by where u(t) is the input and y(t) is the output, is a linear time invariant system.

Hint: Use the result of problem 6. To prove the time invariance property, show that under zero initial conditions if the output of the system to the input, u(t), is y(t), then the output of the system to a delayed version of the input,  u(t-t₁), is  y(t-t₁) for all  t₁  > 0. You need to assume that the value of the input before t₀ is zero.

Problem 8: In DC motor, 

(a) Is the system linear? Justify with theorem that is proved in Problem 7.

(b) Time invariant? Justify with theorem that is proved in Problem 7.

(c) Outline the steps on how you can use Simulink to demonstrate the linearity and the time invariance of the system.

(d) Use Simulink to show the linearity and the time invariance.

(e) What is the order of the system?

(f) What is the rank of the A matrix? Should the rank and the order number in part (e) be the same? Why or why not?

Problem 9: Consider a linear system with input u and output y. Three experiments are performed on this system using the inputs u₁(t), u₂(t), and u₃(t) for t ≥ 0. In each case, the initial state at t = 0,  x(0), is the same. The corresponding observed outputs are y₁(t), y₂(t), and y₃(t).

(a) Which of the following three predictions are true if x(0) ≠ 0? Why?

1) If u₃ = u₁ + u₂, then y₃ = y₁ + y₂.

2) If u₃ = ½(u₁ + u₂), then y₃ = ½(y₁+y₂).

3) If u₃ = u₁ - u₂, then y₃ = y₁ - y₂.

(b) Which of the statements in part (a) are true if x(0) = 0 ? Why?