I really cannot follow what you did because it came out too messed up, ...and in fact you shouldn't be asking this: just multiply the original matrix by what you got and if you get the identity matrix you're done, otherwise you're wrong.
About the decimals: drop them! If you divide by three then write , and not that horrible thing , which is not even a good approximation and it looks childish and,again, pretty ugly. Simple fractions are as numbers as anything else, so befriend them!
No wonder you got all that mess if you jump between decimal and simple fractions...