I'm searching for the name of these sets of matrices:
2-dimensional: $$\quad M_1^{2d}=\left(\begin{matrix}1 &0\\0 &1\end{matrix}\right), M_2^{2d}=\left(\begin{matrix}0 &1\\1 &0\end{matrix}\right)$$
3-dimensional: $$\quad M_1^{3d}=\left(\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right), M_2^{3d}=\left(\begin{matrix}1&0&0\\0&0&1\\0&1&0\end{matrix}\right), M_3^{3d}=\left(\begin{matrix}0&1&0\\1&0&0\\0&0&1\end{matrix}\right)$$ $$M_4^{3d}=\left(\begin{matrix}0&0&1\\0&1&0\\1&0&0\end{matrix}\right), M_5^{3d}=\left(\begin{matrix}0&1&0\\0&0&1\\1&0&0\end{matrix}\right), M_6^{3d}=\left(\begin{matrix}0&0&1\\1&0&0\\0&1&0\end{matrix}\right) $$
It's something like the basis-set of orthogonal matrices, but without signs. I was searching for it, but I'm ending up at orthogonal or unitary matrices, and my matrices are something more specialized.
These are the permutation matrices. They are given this name because each such matrix permutes the entries of a vector under multiplication. For example, $$ \pmatrix{&1\\1\\&&1} \pmatrix{x_1\\x_2\\x_3} = \pmatrix{x_2\\x_1\\x_3} $$ Such matrices are important, for example, in representing graph isomorphisms and in the definition of a reducible matrix, which provides an important connection between matrix-analysis, graph-theory, and the study of Markov chains.