(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.
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Let A be a matrix which is both symmetric and skew symmetric. so AT=A and AT= -A so A =- A that implies 2A =zero matrix that implies A is a zero matrix
A skew symmetric matrix is a square matrix which satisfy, Aij=-Aji or A=-At
The eigen values of a real symmetric matrix are all real.
a square matrix that is equal to its transpose
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.