Is there a matrix that can be used to find the transpose of a matrix? Let $A$ be a general $n\times n$ invertible matrix. Let $T^A$ be the "transposer" matrix i.e. $T^A A = A'$. (Does that $T^A$ multiplied by $A$ equal the transpose of $A$?) Then does $T^A$ depend on the matrix $A$: is $T^A = T^B$ for all invertible Matrices? Prove your claim.
So can you take a matrix times the given matrix in order to find its transpose? If so, is that a general form that can be used for all invertible matrices? 
I'm guessing it does not, but I am not totally sure or know how to go about proving that and I cannot find anything online about it.

Even in the simple case of $2\times 2$ matrices such a general transposer matrix does not exist. 
Note that in order to keep the diagonal terms unchanged you need the so called transposer to be the  identity matrix which in general does not change the matrix to its transpose.
