/sci/ - Science & Math

Hi, I'm currently trying to prove that G (hereby defined as the set which contains all 2x2 matrices) is a group under matrix multiplication.

Associativity and existence of the identity are done, but I'm stuck on the inverse element. I know it's just the inverse matrix, so that

<span class="math"> A \cdot A^{-1} = id [/spoiler]

(where A, its inverse and id are in G, and id is the unit matrix)

I'm sure it's really simple, but if I want to solve the equation above, how can I ensure that the variables encircled in the image always yield 1 when used in the multiplication? I'm not seeing it.
Thanks for any help!