In openGl ES and the World of 3D - why use the invers matrix?

The thing is that I dont have any intuition to, why it is used, therefore please correct me:

As fare as I understand, it is used in shaders - and can help you to figure out the opposite direction of the normals?

Invers in ordinary numbers is like; The product of a number and its multiplicative inverse is 1. Observe that 3/5 * 5/3 = 1. In a matrix this will give you the Identity Matrix, which is the base coordinate system or the orion of the World space - right. But the invers is - some other coordinate system?

You can use the transpose(Row-major order to Column-major order) of a square matrix to find the inverted matrix, as calculating the invers is process heavy - and the transpose is giving you the inverted matrix as a bi product?

Again, I am looking for getting some intuition of this - and therefore be able to use it as intended. Thank you for any reply that will guide me in the right direction.

Regards