Matrix Permutation

Question

Show: If(P1 P2)is a permutation matrix, then(P2 P1)is also a permutation matrix.

Answer 1

Hi Bernard, the matrix (P2 P1) is obtained from X = (P1 P2) by interchanging columns of X. So there is a sequence of elementary matrices E1,..., Es such that the product XE1•••Es = (P1 P2). Each elementary matrix Ej represents the interchange of two columns of the identity matrix, so they are permutation matrices. Since the product of permutation matrices is a permutation matrix, if X is a permuation matrix, so is XE1•••Es. Thus (P2 P1) is a permutation matrix.

Answer 2

For n=2, the two permutation matrices are

 

 

(1   0)

(0   1)   

 

AND

 

(0   1)

(1   0)

So P1 and P2 are column vectors, either
(0)
(1)

or (1)
(0)

Either way, the identity holds