Just checking I'm following what you're after: if you call your original matrix A, and your inverse is A^-1, you want A^-1A = I?

If so, your inverse is

-a/(-a^2 - 1) -1/(-a^2 - 1)

-1/(-a^2 - 1) a/(-a^2 - 1)

If your original matrix was its own inverse, then AA=I, but in your case AA =

a^2 + 1 0

0 1 + a^2

Which is (a^2 + 1)I, but not I itself.