Full Rook Matrix is Invertible

Theorem
A full rook matrix is invertible.

Proof
Let $\mathbf A$ be a full rook matrix.

By definition, $\mathbf A$ is an instance of a permutation matrix.

By Determinant of Permutation Matrix, it follows that $\det \mathbf A = \pm 1$.

By Matrix is Invertible iff Determinant has Multiplicative Inverse:
 * $\mathbf A$ is invertible.