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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $A$ be an $n \\times n$ [[square matrix|0JXZKX526R]] with entries in $\\Z$.\nThe matrix $A$ is *unimodular* when its [[determinant|00KDZRW4C0]] is $1$ or $-1$.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 2 \u0026 3 \\\\ 1 \u0026 2 \\end{pmatrix}$ is unimodular, because its determinant is $4 - 3 = 1$.\nNon-example: $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$ is not unimodular, because its determinant is $2$; its inverse $\\begin{pmatrix} 1/2 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$ is not an integer matrix.\nAn integer square matrix is unimodular exactly when it is [[invertible|0EB9N0S33C]] and its inverse has integer entries.\nThis follows from $\\det(A)\\det(A^{-1}) = 1$ and from the formula of the inverse by the adjugate.\nMore generally, over a commutative ring $R$, a square matrix is called unimodular when its determinant is a [[unit|0XVEZDE9J8]] of $R$.\n",
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