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(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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Prove that a matrix which is both symmetric as well as skew symmetric is a null matrix?
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
What is a skew symmetric matrix?
A skew symmetric matrix is a square matrix which satisfy, Aij=-Aji or A=-At
What is the eigen value of a real symmetric matrix?
The eigen values of a real symmetric matrix are all real.
What is a symmetric matrix?
a square matrix that is equal to its transpose
Why are unrestrained global stiffness matrix singular?
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.
