Finding the unknown matrix in an equation?

Question

so I was wondering how can I find the unknown matrix from an equation, I need to find X

    [-1 2]  X  [1 0]    [-2 -12]
    [ 0 1]     [2 4]  = [1  - 4]

so I started with assigning each matrix with a letter A, B, C respectively, and I have X I need to find, and I have managed to find the inverses, of all three if Its needed which I guess it is, so now what do I do next? thanks for any help :)

Answer 1

$$AXB=C$$ $$XB=A^{-1}AXB=A^{-1}C$$ $$X=XBB^{-1}=A^{-1}CB^{-1}$$

Answer 2

Use the superoperator formalism: $$ AXB=C \mapsto (\hat {B}^T\otimes \hat A)\text{vec} X = \hat C $$ (see here for a definition of $\text{vec}(X)\;$ and here for more information: Kronecker product)