## 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 ?