Suppose I have a certain matrix equation 1
$$A_{2\times1}=M_{2\times 2}B_{2\times 1}$$
and matrix equation 2
$$Ka_{2\times1}=Mr_{2\times2}Kb_{2\times1}$$
And suppose $A^T\cdot Ka=B^T\cdot Kb=1$
Then, I can write $$A^T_{1\times2}=B^T_{1\times 2}\cdot M^T_{2\times2}$$ Then multiplying this to equation 2 I get $$A^T_{1\times2}\cdot Ka_{2\times 1}=B^T_{1\times2}\cdot M^T_{2\times 2}\cdot Mr_{2\times2}\cdot Kb_{2\times1}=1$$
From here, I need to prove that $M^T_{2\times2}=Mr_{2\times2}^{-1}$
But, since matrix multiplication is not associative, I guess that I can't group $B^T$ and $Kb$ together. So, how do I proceed? Your help will be appreciated :)