Mrs. Rogers is only the second person to die on the island. The characters are first beginning to make the connection between the the ten statues, the poem, and the ten of them. They have all been informed of the crimes each of them has allegedly committed. Mrs. Rogers was very upset when she was accused, with her husband, of killing their employer. As a group, the travelers decide Mrs. Roger's death could be one of two possibilities. Dr. Armstrong lists the possibilities in a discussion with Mr. Lombard as:
"Rogers killed her because he was afraid she would give the show away. Second possibility: she lost her nerve and took an easy way out." (pg 115)
Lombard considers the possibility of suicide EXCEPT:
"It could have been --- yes---if it hadn't been for Marston's death. Two suicides within twelve hours is a little too much to swallow!" (pg 115)
Mr. Lombard admits that individually each character could have committed suicide, but both of them committing suicide would be incredulous.
"I could believe in Anthony's suicide (with difficulty) if it weren't for Mrs. Rogers. I could believe in Mrs. Rogers' suicide (easily) if it weren't for Anthony Marston. I can believe that Rogers put his wife out of the way --- if it were not for the unexpected death of Anthony Marston. But what we need is a theory to explain two deaths following rapidly on each other." (pg 117)
So, Lombard doesn't believe Mrs. Rogers killed herself because of the death of Anthony Marston, and the fact that two people died in such rapid succession.