Earlier quoted context omitted.
No. Column scrambling doesn't affect row ordering and row scrambling doesn't affect column ordering. They are literally orthogonal. You can't tell, from a scrambled matrix, whether the rows or the columns were scrambled first. Here's a more explicit breakdown: (0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2) Column scrambling: swap 1&2: (0,0) (2,0) (1,0) (0,1) (2,1) (1,1) (0,2) (2,2) (1,2) Now row scrambling: sw…
Another way to think of it is that permuting rows corresponds to multiplying a matrix on the left by a permutation matrix, and permuting columns, on the right. Since matrix multiplication is associative, they commute!
Yet another way to think of it is that a matrix is an encoding of a linear (or affine) function, matrix multiplication is composition of such functions, and since composition is associative, they commute. :)