I never quite understood why her pathetic attempts to defend Rand in that scene with Gawyn were phrased the way they were. She was there when he heard that Morgase had (allegedly) been murdered by Rahvin, and decided to attack to avenge Elayne's mother. That's all she had to say. Instead, she leaves open the possibility that Rand did kill Morgase by saying she couldn't swear to knowing for certain that it wasn't Rand. She goes on to say she will try to prove Rand's innocence, something she has no intention of ever doing. Of course, Gawyn is completely unreasonable about this whole thing, especially considering he knows after Dumai's Wells that his sister loves the man who supposedly killed their mother, but Egwene does nothing to set him straight here.