math question about the matrix analysis

During the structural analysis we allways get a symmetric matrix. But what makes me wonder is from my linear algebra course I remember that not all matrices are invertible. And obviously not all symmetric matrices are invertible ([1 1; 1 1]).

So what is it that makes all the matrices we get in the structural analysis invertible? I mean, physically if it wasnt invertible it would mean that we could get more sollutions, because the system Ax=b isn't unique, and is this really that unlikely? but I would mostly like to know if there is a mathematical answer to this question.

Please help.:)