The correct answer is: A. Stable and causal.
A system is said to be stable if its output does not grow unbounded in response to a bounded input. A system is said to be causal if its output depends only on the present and past inputs, not on future inputs.
The impulse response of a system is the response of the system to a unit impulse input. The impulse response of a stable system is always non-negative. The impulse response of a causal system is always zero for negative time.
The impulse response of the system in the question is $h(n) = 2n u(n – 2)$. This is a non-negative function, which means that the system is stable. It is also zero for negative time, which means that the system is causal. Therefore, the system is stable and causal.
Here is a brief explanation of each option:
- Option A: Stable and causal. This is the correct answer. The system is stable because its impulse response is non-negative. The system is causal because its impulse response is zero for negative time.
- Option B: Causal but not stable. This is not the correct answer. The system is causal because its impulse response is zero for negative time. However, the system is not stable because its impulse response is not non-negative.
- Option C: Stable but not causal. This is not the correct answer. The system is stable because its impulse response is non-negative. However, the system is not causal because its impulse response is not zero for negative time.
- Option D: Unstable and noncausal. This is not the correct answer. The system is stable because its impulse response is non-negative. The system is also causal because its impulse response is zero for negative time.