Bingo. Most Matrix operations require you have numbers as entries; not even fractions work. I think this is on a lot of peoples' wish lists.
b) So I punted and just entered the transpose myself, but now when I have WebWork display the matrices so the students can enter them, the ones with variables in them don't display as a matrix, but simply as a single blank. See attached image.
Have you tried taking out the quotation marks? Unless WeBWorK went through a major revision recently, you shouldn't need them.
c) Is there a better way to do the 'impossible' case?
I'm going to punt on that one; I've never done essay questions before in WeBWorK.