A permutation matrix $A$ is a nonsingular square matrix in which each row has exactly one entry = $1$, the other entries being all zeros. If $A$ is an $n×n$ permutation matrix, what are the possible values of determinant of $A$?
——-I think any such given matrix is a row column interchange operation of the Identity matrix.since change of row or column once change the sign only and the identity matrix has determinant $1$ so the answer will be $+1$ or $-1$.Am I right?
Does the logic is correct?
Best Answer
(Just so this question has an answer.)
As fkraiem points out, a permutation matrix is a matrix with precisely one $1$ in each row and precisely one $1$ in each column, and zeroes elsewhere.
Your argument is correct as is your conclusion. Furthermore, $n\times n$ permutation matrices are in one-to-one correspondence with $S_n$, the group of permutations on $n$ elements (hence the name). The determinant of a permutation matrix is the sign of the corresponding permutation.