I've been given a equation where I need to find $C$ (matrix) where I know what $A$ and $B$ are (also matrices), the equation is :
$$ 13I= -0.2*A*C+B $$
I want to canonize $B$ with elementary row operations to make it easier - can I do that? if I can does it affect the value of $B$? if so, how can I negate that?
It would help to know the sizes of $A,B$, and $C$. Assuming they are all square matrices, then you can rewrite your equation as $-5(13I - B) = AC$. Then you can solve this system simultaneously by considering the augmented matrix $$[A \vert 5B - 65I].$$ Then you need to do Gaussian elimination (row operations) to put this into reduced row echelon form. If this system only has one solution, then the reduced row echelon form will be of the following form $$[I \vert C].$$ Otherwise, there will be some free parameters or there is no solution.