If $A,B$ and $C$ are matrices and $A = BC$, then what is $B$ in terms of $A$ and $C$?

Question

Will the matrix inverse of C be pre-multiplied or post-multiplied with A?

Meaning:

if A = BC, then 

is B = A * C^-1

or

B = C^-1 * A

How does this change when a is a vector, b is also a vector but C is a product compatible matrix for the vector-matrix multiplication to happen in B and C.

Answer 1

Since the matrix product is non-commutative, you can't do $C^{-1}BC$ and expect it to simplify. You have therefore to multiply by $C^{-1}$ on the right, giving the first option.