The old man is even more of a mystery than the narrator, partly because we only see him through the narrator's skewed perspective. We know he has money (the narrator shows the old man's "treasures" to the police). We also know he has a blue eye that the narrator is afraid of, and which fits the description of a corneal ulcer. We know he's old, and that he's a fairly sound sleeper. Not much meat for a character study, though. Luckily, we are given some hints to work from.
According to the narrator, the old man suspects nothing because the narrator was super duper nice to him the week before he killed him. We can't prove the old man wasn't suspicious, but because he leaves his bedroom door unlocked we can assume it. We know the man isn't naturally trusting – he's afraid of robbers. But, it seems he does trust the narrator enough to give him the run of the house while he sleeps. Unless the old man is a poor judge of character, or senile, his trust suggests that the narrator really is capable of acting sanely.
Nothing the narrator tells us about the old man fits our idea of "madness" or "insanity," but the old man does fit neatly into the narrator's definition of madness: 1) "destroyed" or "dulled" senses; 2) "Madmen know nothing" (2).
Sounds like the old man, right? His senses are definitely dulled – he only hears the narrator on the eighth night. He doesn't seem to have the slightest idea what's going on around him and is incapable of defending himself. Perhaps the narrator is slyly hinting that he thinks the old man is "mad." This makes us wonder if the old man was very senile, dependant on the narrator's care. If so, this adds a new dimension to the creepiness and puts the narrator in an even more negative light.
We know that at least one neighbor is suspicious of the goings on in the house of the old man and the narrator. Otherwise, he or she would not have been so quick to call the cops after hearing a little scream, and wouldn't have been able to convince the powers that be to send not one or two, but three policemen. We don't know if this suspicion is directed toward the old man or toward the narrator or both. But, it's possible that the narrator wasn't the only one afraid of the old man's eye. The old man could be an alienated figure both in and out of the home, and thus the narrator's murder of him could be symbolic of prejudices and abuses that stem from physical "difference."The Old Man Timeline