I am trying to use an inversed matrix to solve system of equations, however, I have encountered a problem that is making me confused.
General Step for using an inversed matrix to solve system of equations:
$AX=B$ (Step 1)
$A^{-1}AX=A^{-1}B$ (Step 2)
$X=A^{-1}B$ (Step 3)
Specifically, for step 2, I know the fact that matrix multiplication is not commutative (So $A^{-1}B ≠ BA^{-1}$ ). That is why I would like to ask:
Why, when we multiply $A^{-1}$ on the left of equation, the right side is $A^{-1}B$ but not $BA^{-1}$ ?
And for future, how can I know the position of the inverse matrix, when I multiply it by both side?
Ex: For AB=ED,
Should $A^{-1}AB=EDA^{-1}$?
Or $A^{-1}AB=EA^{-1}D$?
Or $A^{-1}AB=A^{-1}ED$?