Why $AB=X$ is not a valid linear system?

This might be a dumb question but I’ve been thinking about it all day. I’m currently reading on my own a book about the fundamentals of electronic circuits and I’m in the chapter of node and mesh analysis. Because this is not the Electrical Engineering forum I will go straight to the mathematical question.

It is a fact that: $ R*I = V $

Also, conductance is the inverse of resistance : $ G = 1/R$ , so we can say that : $ G*V = I$

So we have a $ N*N$ linear system of $ G*V = I$ where:

  • G an $ N*N$ matrix (we know every element of it)
  • V is a $ N*1$ vector (these are the variables)
  • I is a $ N*1$ vector (we also know every element of it)

This is a usual $ AX=B$ linear system where we choose the Inverse Matrix ,Cramer or Gaussian elimination method to solve it.

But if I make a new $ R$ matrix based on $ R = 1/G$ and then have a $ N*N$ linear system(?) of $ R*I = V $ where:

  • R an $ N*N$ matrix (we know every element of it)
  • I is a $ N*1$ vector (we also know every element of it)
  • V is a $ N*1$ vector (these are the variables)

which is of form $ AB=X$ then if I make a matrix multiplication I will not get the same $ V$ vector.

I know that in matrices $ AB \ne BA$ but that just doesn’t convince me. Can someone give me a solid proof of why this is wrong ?