For two matrices of dimensions $m \times n$ and $n \times k$, define $C=A\odot B$ to be the matrix with entries
$$C_{ij}=\max_{k=1}^n A_{ik} + B_{kj}$$.
Given an $n \times n$ matrix $A$, is there an $n \times n$ matrix $E$ such that $A\odot E=A$?
Also, given an $n \times n$ matrix $A$, is there an $n \times n$ matrix $F$ such that $A \odot F=E$ where $E$ is as above?
My initial thoughts: After playing around with the matrices, I believe that $E$ can be the $n \times n$ matrix in which the diagonal consists completely of $0$'s and the rest of the matrix is filled by the smallest value of $A_{ij}-B_{ij}$. Would this matrix $E$ apply for all situations?
