Consider the set of matrices that satisfy the property
Explicitly, in matrix form, the elements are arranged as follows
The matrix is sometimes referred to as the exchange matrix.
Using the exchange matrix the above set of matrices can also be conveniently defined as
Proposition. The set of matrices forms a group.
Proof. It can be easily shown that the identity is in the group, i.e. . Also the set is closed under multiplication, i.e. if and then . To see why, consider
If , then
Therefore, . This completes the proof that forms a group.