What is your field? Remember, a matrix is a linear map between two vector spaces over a specific field...Basically, are you working over , or ?
I am asking you this because the field does matter; let be the identity matrix, and assume we are working over . Then , as (not) required.
Thus, your field does matter so you'll need to be careful and you will need to use your field. However, your second approach looks promising. Essentially, you have that . So, hit your matrix with a stick - stick in variables for your 9 entries and see what happens when you take its inverse or negate it (there will be a more subtle way, but I can't think of it at the moment!).
EDIT: Note that exists because so . So splitting into makes sense...(I'm sure you know this, but I thought I should probably point it out anyway).