TITLE: Can you transpose a matrix using matrix multiplication?
QUESTION [7 upvotes]: Say you have a matrix A = \begin{bmatrix}a&b\\c&d\end{bmatrix}
and I want it to look like $A^T$ = \begin{bmatrix}c&a\\d&b\end{bmatrix}
Can this be done via matrix multiplication? Something like a matrix T such that $T*A = A^T$.

REPLY [15 votes]: If there were such a $T$, we would have that $T = T \times I = I^T = I$, where $I$ is the identity matrix. But then it would follow that $A = I \times A = T \times A = A^T$ for all matrices $A$; i.e., that all matrices are their own transposes. As this is not true, we conclude there cannot be any such $T$ as desired.