Generalized permutation matrix

In matrix theory, a generalized permutation matrix is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column.

An example of generalized permutation matrix is

${\displaystyle {\begin{bmatrix}0&0&3&0\\0&-2&0&0\\1&0&0&0\\0&0&0&1\end{bmatrix}}}$

An interesting theorem states the following:

If a nonsingular matrix and its inverse are both nonnegative matrices (i.e. matrices with nonnegative entries), then the matrix is a generalized permutation matrix.