Full Rook Matrix is Invertible
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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.
$\blacksquare$
Sources
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.4$: Multiplication and inverses: Ponderable $2.2.2 \ \text{(a)}$