The state variable description of an LTI system is given

The state variable description of an LTI system is given by

where y is the output and u is the input. The system is controllable for

1. a₁ ≠ 0, a₂ = 0, a₃ ≠ 0
2. a₁ = 0, a₂ ≠ 0, a₃ ≠ 0
3. a₁ = 0, a₂ ≠ 0, a₃ = 0
4. a₁ ≠ 0, a₂ ≠ 0, a₃ = 0

Q_c = [B AB A²B]

If rank of Q_c = 3 = order of matrix, then
Q_c is controllable , then |Q_c | ≠ 0