As you answer this question, you need to remember that there is no right or wrong answer. The story is not written to solve a murder mystery. The author's purpose is to demonstrate that everyone has his or her own reality and sees the world through the lens of that reality. In fact, eyewitness testimony is notoriously unreliable for this reason.
There is evidence to support that the wife or the bandit murdered the victim or that he killed himself. A close reading of the story will allow you to find details that support any one of the three possibilities. Choose one of the three, and look at the evidence. What was the person's motive? Was there opportunity to commit the murder (or suicide)? What was the weapon? Did the person have the physical capability to do this? These are the questions you must answer to establish an argument for one of the possible killers.