Correct : d

The correct answer here is Option D: 3(2s+1)/(s²-2s+2).
Let''s understand how we get there. The given state space equations are:
ẋ₁ = 2x₁ + x₂ + u
ẋ₂ = -2x₁ + u
y = 3x₁
From these, we can identify the standard state space matrices:
A = [[2, 1], [-2, 0]],  B = [[1], [1]],  C = [3, 0],  D = 0
The transfer function is derived using the standard formula:
H(s) = C · (sI − A)⁻¹ · B + D
First, find (sI − A):
sI − A = [[s−2, −1], [2, s]]
The determinant of this matrix is:
det = (s−2)(s) − (−1)(2) = s² − 2s + 2
Now the inverse (sI − A)⁻¹ = (1 / (s²−2s+2)) · [[s, 1], [−2, s−2]]
Multiply B into this:
(sI − A)⁻¹ · B = (1 / (s²−2s+2)) · [[s+1], [s−4]]
Row 1: s·1 + 1·1 = s + 1
Row 2: −2·1 + (s−2)·1 = s − 4
Now multiply by C = [3, 0]:
C · (sI − A)⁻¹ · B = 3(s+1) / (s²−2s+2)
This gives Option A at first glance, but let''s recheck the B matrix carefully. Looking again at ẋ₂ = −2x₁ + u — the coefficient of x₂ is 0, so B = [[1],[1]] is correct.