We have an elementary operations of swapping rows and columns of the matrix, i.e. given $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ we can swap rows by $$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} c & d \\ a & b \end{bmatrix}$$
My interest is to swap two adjacent elements in the matrix, and I hopped it could be achieved by some matrix multiplication.
Unfortunately, this is not the case. We expect such transformations to be reversible, but then it is easy to move elements off the main diagonal of the identity matrix to make it singular.
Are there any other elementary operations on matrices, except row/column swap, which shuffles elements of the matrix by the means of matrix multiplication?