By the way, this process can be used for 3 x 3's and bigger. (You can only find the inverse of a square matrix.) BUT, the process gets MUCH stickier! This is really intended for computers.
Here's the general idea:
Here's the double matrix:
Use row operations to get
With a lot of VERY careful work, you CAN do it. (No, you don't have to do it right now.)