Let $ABC = D$ where $B$ and $D$ are symmetrical matrices. However their [rows x columns] values are not same. For example, $B$ is 2x2 and $D$ is 3x3 a matrix. Clearly, in this case, $A$ has to be a 3x2 matrix and $C$ must be a 2x3 matrix.
Prove or disprove that this holds iff $A^T = C$ i.e. if $A$ and $C$ are transpose to each other then $D$ is symmetrical and if $B$ and $D$ are symmetrical then $A$ and $C$ are transpose to each other. I have tested few cases on computer and it seems to be correct. But I am not sure about 'if and only if' part. Does this hold if $B$ and $D$ are positive definite? Any comment about the nature of $A$ and $C$ is welcome. A chocolate for a correct proof, a cup of coffee otherwise!
EDIT : There was a serious flaw in original problem. Corrected now. Thanks to Hardmath and Willie.