How prove this matrix equation $AXB=C$ has a solution

Question

Qustion:

if Matrix equation $AY = C$ and $ZB = C$ has solution,

show that:

the equation of $AXB= C$ has solution

This problem is from this PDF(page 3) problem 2 :http://wenku.baidu.com/view/d6625d1cff00bed5b9f31dba.html

My idea:we have $$\left(A\bigotimes I\right)\overline{Y}=\overline{C}$$ $$\left(I\bigotimes B^T\right)\overline{Z}=\overline{C}$$ we only prove $$\left(A\bigotimes B^T\right)\overline{X}=\overline{C}$$ has solution.

since $$\left(A\bigotimes I\right)\left(I\bigotimes B^T\right)=A\bigotimes B^T$$ then ony prove follow $$I_{m}\left(A\bigotimes I\right)\bigcap I_{m}\left(I\bigotimes B^T\right)\subset I_{m}\left(A\bigotimes B^T\right)$$ My idea is wrong? If is true,and follow How works? Thank you

and maybe have other methods.Thank you very much!

Answer 1

I will assume that your matrices are over a regular ring (maybe even over a field). Then each of these is regular. By this I mean for $A$ there is matrix $A^-$ such that $AA^-A=A$.

Then the first two equations you have imply that

$AA^-C=C$ and

$CB^-B=C$.

Now you may like to guess what a solution of $AXB=C$ may be. If not see bellow.

Answer $X=A^- CB^-$.