TITLE: Can the transpose of a matrix be expressed in row/column operations?
QUESTION [6 upvotes]: Suppose that $A$ is a matrix, can we get its transpose, $A^T$, by performing row and/or column operations to $A$?

REPLY [5 votes]: There may be two things you might be asking, and the answer depends on which is the right one.
In the body of your question, you are fixing a matrix $A$ first, and asking if you can get to $A^T$ by a sequence of row and column operations. This is the same as asking if $A$ and $A^T$ are equivalent matrices. Since equivalent matrices must have the same shape, one must require that $A$ be square, but given that the answer is "yes": two matrices of the same shape are equivalent if and only if they have the same rank, and it is well known that $A$ and $A^T$ always have the same rank (a result also known as "row rank equals column rank").
However, your title asks "Can the transpose of a matrix be expressed in row/column operations?", which means the oprations performed should not depend on the matrix itself, but rather work for all matrices at once. Here the answer is a resounding "no", such an expression is not possible for square matrices of size $n>1$. Here's one reason why the is not possible. Any combination of a fixed mixture of row and column operations, results in a transformation $M\mapsto RMC$ where $R$ and $C$ are invertible matrices determined by the row respectively column operations. Now if one wants $RMC=M^T$ for all $M$, then taking $M=I$ gives us that $RC=I$, in other words $C=R^{-1}$. We can then write the requirement $RMC=M^T$ for all $M$ as $RM=M^TR$ for all $M$. If you write out what that means for $M=E_{i,j}$, you will that all entries of $R$ in the column (number $i$) of $R_{j,i}$ other than that entry itself have to be $0$, as well as all entries of $R$ in the row (number $i$) of $R_{i,j}$ other than that entry itself (in addition those two remaining entries must be equal, but I won't use that). Since $i,j$ can be chosen arbitrarily in the range $1,\ldots,n\geq2$, this can only hold if all entries of $R$ are zero, but that is in contradiction with the requirement that $R$ be invertible. Conclusion: such matrices $R,C$ do not exist.