## Permutation Matrix

A Matrix obtained by permuting the th and th rows of the Identity Matrix with . Every row and column therefore contain precisely a single 1, and every permutation corresponds to a unique permutation matrix. The matrix is nonsingular, so the Determinant is always Nonzero. It satisfies

where I is the Identity Matrix. Applying to another Matrix, gives A with the th and th rows interchanged, and gives A with the th and th columns interchanged.

Interpreting the 1s in an permutation matrix as Rooks gives an allowable configuration of nonattacking Rooks on an Chessboard.