Is matrix transpose a linear transformation? This was the question posed to me. Does there exist a matrix $A$ for which $AM$ = $M^T$ for every $M$. The answer to this is obviously no as I can vary the dimension of $M$. But now this lead me to think , if I take , lets say only $2\times2$ matrix  into consideration. Now for a matrix $M$, $A=M^TM^{-1}$ so $A$ is not fixed and depends on $M$, but the operation follows all conditions of a linear transformation and I had read that any linear transformation can be represented as a matrix. So is the last statement wrong or my argument flawed?

While current answers clearly indicate why the reasoning is not correct, I would like to add one point that has not been mentioned so far. In trying to define $A=M^TM^{-1}$, you are making a matrix $A$ that depends on the argument $M$ that the map is being applied to. That can never be right. While describing a linear transformation as a map involves, like for any map, an expression that gives the result in terms of the argument (as here $M\mapsto M^T$), the matrix representing the linear map must by definition contain constant entries, values that do not depend on the argument (here $M$) the linear map is  potentially going to be applied to. This is similar to the coefficients of a polynomial function of$~x$: these coefficient by definition cannot involve$~x$. See also this answer to a related confused question.
Of course this takes nothing away from the other arguments, mostly that $A$ is not the right type of matrix to be considered the matrix representing the linear map $M\mapsto M^T$ (and not to mention the problem you would have for the case of non-square matrices, for which $M\mapsto M^T$ is still a linear transformation).
