Can You Completely Permute The Elements Of A Matrix By Applying Permutation Matrices?
Answer : It is not generally possible to do so. For a concrete example, we know that there can exist no permutation matrices P , Q P,Q P , Q such that P\pmatrix{1&2\\2&1}Q = \pmatrix{2&1\\2&1} If such a P P P and Q Q Q existed, then both matrices would necessarily have the same rank. Let me add one more argument: For n ≥ 2 n \ge 2 n ≥ 2 : Suppose the entries in the n × n n \times n n × n matrix A A A are all distinct. Then there are ( n 2 ) ! (n^2)! ( n 2 )! distinct permutations of A A A . There are n ! n! n ! row-permutations of A A A (generated by premultiplication by various permutation matrices), and n ! n! n ! col-permutations of A A A (generated by post-multiplication by permutation matrices). If we consider all expressions of the form R A C RAC R A C where R R R and C C C each range independently over all n ! n! n ! permutation matrices, we get at most ( n ! ) 2 (n!)^2 ( n ! ) 2 possible results. But for n > 1 n > 1 n > 1...