A permutation matrix is a matrix obtained by permuting the rows of an 
identity matrix according to some permutation of
the numbers 1 to 
.
Every row and column therefore contains precisely a single 1 with 0s everywhere else,
and every permutation corresponds to a unique permutation
matrix. There are therefore 
permutation matrices of size 
, where 
is a factorial.
The permutation matrices of order two are given by
![[1 0; 0 1],[0 1; 1 0]](/images/equations/PermutationMatrix/NumberedEquation1.svg) |
(1)
|
and of order three are given by
![[1 0 0; 0 1 0; 0 0 1],[1 0 0; 0 0 1; 0 1 0],[0 1 0; 1 0 0; 0 0 1],[0 1 0; 0 0 1; 1 0 0],
[0 0 1; 1 0 0; 0 1 0],[0 0 1; 0 1 0; 1 0 0].](/images/equations/PermutationMatrix/NumberedEquation2.svg) |
(2)
|
A permutation matrix is nonsingular, and the determinant is always 
.
In addition, a permutation matrix 
satisfies
 |
(3)
|
where 
is a transpose and 
is the identity matrix.
Applied to a matrix 
, 
gives 
with rows interchanged according to the permutation vector
,
and 
gives 
with the columns interchanged according to the given permutation vector.
Interpreting the 1s in an 
permutation matrix as rooks
gives an allowable configuration of nonattacking rooks
on an 
chessboard. However, the permutation matrices provide
only a subset of possible solutions.
