I have an student who is telling me that if I have a matrix $A$, and a final matrix $b$, that multiplying $A*b$ will give me the solution of systems matrix $x$. I know that a system of linear equations can be represented as:
$$Ax = b$$
What this fellow student is telling me is that if we generate a random $A$ and a random $b$, that if we do the following, we can get $x$:
$$Ab = x$$
Algebraically, this just seems blatantly wrong; not to mention logically as well, otherwise what would be the point of Gaussian Elimination??
Now the problem set:
- Take user input as $N$
- Generate a random matrix $A$ such that it is $NxN$
- Generate a random matrix $sol$ such that it is $Nx1$
- Multiply $A*sol$ to get $b$ such that it is $Nx1$
- Store $A$ to file and $sol$ to file
- Read files into new program and solve via Gaussian Elimination
- Compare results and display normalized error
It is my belief that $A$ = $A$ in $Ax=b$, and $sol$ = $x$ in $Ax=b$. The other student believes that $sol$ = $b$ in $Ax=b$ which to me makes no sense.
My position is that the instruction set has an error in the last 3 steps. If we store $A$ and $sol$ and use Gaussian Elimination, we cannot do that because we already have the $Ax$ piece of the system $Ax=b$. What we should also be storing is $b$, read in all the files, perform Gaussian Elimination on $A$ and $b$ to find a new $x$ and compare our $x$ to our original $sol$.
Looking for confirmation of my assumption or that I am clearly not grasping what I thought was a simple linear algebra problem...