Why does not the following matrix transformation seem to be possible?

Basically I have 3 matrices, and matrix 3 is the transpose of matrix 1. I wonder why it is impossible to find a matrix transformation A such that A*M1=M3 (that works for all the transpose matrix pairs).

I initially thought applying a reflection matrix around y=x would be helpful, but upon trying it turned out just to be matrix 2, which seems to have exactly the same shape as matrix 1.

I don't have very profound mathematical knowledge so I'd appreciate simpler answers, thanks!

• M1: https://i.stack.imgur.com/zsNyo.png
• M2: https://i.stack.imgur.com/l7pX7.png
• M3: https://i.stack.imgur.com/thmAA.png

So you want a matrix $A$ such that for any matrix $M, AM = M^T$? But that means it should work for $M = I$. That is, $$AI = I^T = I$$ Since $AI = A$, that means $A = I$. However $IM = M^T$ is only true for symmetric matrices. Thus you cannot transpose all matrices simply by multiplying by a fixed matrix.