What use is the inverse matrix?
I would not use it to solve linear systems but there must be some concrete or real life applications where it is used.
I don't want to tell this myself but there are many practical uses one of which I'm going to show you.
Application of Invertible Matrices: Coding
Please click on the above link to see for yourself.
It is a sort of meta-mathematical statement that in applications in which you have "Ax= y" typically the "A" depends on "situational" properties while the right hand side, y, depends upon specific values. In other words, it is not uncommon to have applications in which you have to solve many such equations with one matrix "A" and many different values of "y". Finding the inverse of A makes it possible to multiply all the various values of y by $\displaystyle A^{-1}$.
In terms of "pure" mathematics, what is important, of course, is knowing that the inverse of a matrix exists and what properties are implied by that.