Linear Simultaneous Eqns - Gauss Elim Problem

[Fjolvar]

Hello, I'm stuck on a simultaneous eqns problem. From what I can see it seems the easiest way would be to get the matrix into row echelon form, but I'm not sure if another way would be better. I can see a pattern here but not sure what it means. I attached the problem to the page. Any help would be greatly appreciated.

 

[malicx]

Hello, I'm stuck on a simultaneous eqns problem. From what I can see it seems the easiest way would be to get the matrix into row echelon form, but I'm not sure if another way would be better. I can see a pattern here but not sure what it means. I attached the problem to the page. Any help would be greatly appreciated.

It is a symmetric square matrix, so if it is invertible it satisfies A^-1A^T = I.

Take a look here, http://en.wikipedia.org/wiki/Symmetric_matrix.

 

[Fjolvar]

Interesting, how would I use this to solve for the variables X1, X2, X3.. etc?

 

[malicx]

Interesting, how would I use this to solve for the variables X1, X2, X3.. etc?

So, what you really have is [A|B], where B is the column vector of solutions, right? So when you multiply by the matrix you have by its inverse what you really get is [I|A^-1B]. Remember when you right it in the form [A|B], it is really just shorthand saying Ax = B. When A is invertible, you have x = A^-1B. Does that make sense?

 

[Fjolvar]

I do understand how to find X by taking the inverse of A and multiplying by B, however I'm still not quite sure of the significance of this matrix being symmetric. So A = A transpose, does that help us in finding the inverse of A?