The inverse of a matrix can be seen as the opposite linear transformation. For example, a vector is changed by a linear transformation. The inverse of the transform matrix can transform the space, so that the changed v would go back to its original position. Actually, it is not only the vector but all the vectors in the space back to the original places. Notably, if a space is squeezed into lower dimensions (no matter real lower dimension, or a lower dimensional space in higher dimension representations), there is no inverse matrix for it.