A singular matrix is one that has a determinant of zero, and it has no inverse. Global stiffness can mean rigid motion of the body.

You most likely your node numbers mixed up somewhere, double check them.

It is the matrix 1/3

It is the matrix 1/3

It is the matrix 1/3

It is the matrix 1/3

(I'm assuming you're referring to FEM)

The entries of a stiffness matrix are inner products (bilinear forms) of some basis functions. Insofar as you will typically be dealing with symmetric bilinear forms, the stiffness matrix will also be symmetric. In other words, ai,j = <φi,φj> = <φj,φi> = ai,j.

The issue is closely related to so-called "Gramian matrices" which, in addition to symmetry, have other properties desirable in the context of FEM. I've provided links below.

To find the original matrix of an inverted matrix, simply invert it again. Consider

A^-1^-1 = A^1 = A