Computing the inverse of a matrix on the fly by hand is not something that students normally get excited about. That's a pity, but it is a fact of life!
On the other hand, it must be admitted that the computational effort required for this operation is significant if you do it by hand and what is worse the operation tends to generate all sort of arithmetic errors.
Yet, the relationship between the simplex procedure and this operation is important and students must be made aware of that even if they are not taught the subject from first principle.
Don't panic if you have do not know what we are talking about here. We prepared for you a very gentle introduction to the subject. Go there if you need help with the basic notion.
On the other hand, you are welcome to go directly to where the action is!