Definition and example
Let denote the th unit row-vector (the th component is 1 and the others are zeros).
Definition 1. A permutation matrix is defined by two conditions: a) all of its rows are unit row-vectors and b) no two rows are equal. We use the notation
where all rows are different.
Example. In the following matrix we change places of rows in such a way as to obtain the identity in the end. Each transformation changes the sign of the matrix by Property VI.
Since there are three changes and the final matrix is identity, by Axiom 3 .
It is possible to arrive to the identity matrix in more than one way but the result will be the same because the number is fixed. We will not delve into the theory of permutations.