matrix operations with transpose like properties Has any one studied operations on matrices with transpose like properties - for example (A+B)' = A' + B' where ' stands for transpose.  Also ' is its own inverse.  Is there a common name for such operations?
 A: Yes, it is a fairly common notion. See Involution.
A: Transposition is linked to the vast domain of duality : 
let us consider for example a $3 \times 3$ matrix.
$$\begin{bmatrix}1&2&5\\-1&1&2\\3&1&2\end{bmatrix}$$


*

*A classical interpretation of the columns of $A$ is in terms of vectors, 

*A dual interpretation gives to the lines of a matrix  the "status" of coefficients of a plane, for example for the first line the plane with equation $ \ \ 1x+2y+5z=0$, or the family of planes with equations  $ \ \ 1x+2y+5z=k$.  More abstractly, when one considers the function $v \in \mathbb{R}^3 \rightarrow \mathbb{R}$ defined by $l_1:(x,y,z)\rightarrow 1x+2y+5z=\langle l_1,v \rangle$ (the last equality being a definition) where $v$ is the vector with components $x,y,z$), using a notation very similar to a dot product. 
$l_1$ is called a "linear form" (acting on vectors to give a number). An illustrative example :
In the product of two matrices, the first one can be considered as line gathering", the second as a "column gathering", and the resulting product is 
$$\begin{bmatrix}l_1\\l_2\\l_3\end{bmatrix}\begin{bmatrix}v_1&v_2&v_3\end{bmatrix}=
\begin{bmatrix}\langle l_1,v_1 \rangle&\langle l_1,v_2 \rangle&\langle l_1,v_3 \rangle\\\langle l_2,v_1 \rangle&\langle l_2,v_2 \rangle&\langle l_2,v_3 \rangle\\\langle l_3,v_1 \rangle&\langle l_3,v_2 \rangle&\langle l_3,v_3 \rangle\end{bmatrix}$$ 
Numerically, you will have no difference, but the interpretation of matrices' lines as "linear forms" is one of the iceberg's tops of the very fruitful domain of duality.   
