Unimodular matrix

Remarks

Example: $\begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$ is unimodular, because its determinant is $4 - 3 = 1$. Non-example: $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ is not unimodular, because its determinant is $2$; its inverse $\begin{pmatrix} 1/2 & 0 \\ 0 & 1 \end{pmatrix}$ is not an integer matrix. An integer square matrix is unimodular exactly when it is invertible and its inverse has integer entries. This follows from $\det(A)\det(A^{-1}) = 1$ and from the formula of the inverse by the adjugate. More generally, over a commutative ring $R$, a square matrix is called unimodular when its determinant is a unit of $R$.