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  "body": "\n## Summary\n\nA unimodular matrix is a square matrix with integer entries and [[determinant|00KDZRW4C0]] $+1$ or $-1$.\nThese are exactly the integer matrices whose inverse also has integer entries.\nThey form the group $\\mathrm{GL}_n(\\Z)$, the [[general linear group|0FCHE902D7]] over the integers.\nUnimodular matrices describe the changes of basis of the lattice $\\Z^n$, and they appear in the Smith and Hermite normal forms.\n\n## Explanations\n\n### L1 human\n\nFor real matrices, any nonzero determinant allows an inverse.\nFor integer matrices, an inverse with integer entries needs more: the determinant must be $1$ or $-1$.\nFor example, $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ has determinant $1$ and inverse $\\begin{pmatrix} 1 \u0026 -1 \\\\ -1 \u0026 2 \\end{pmatrix}$.\nSuch a matrix maps the grid of integer points of the plane onto itself, one to one.\n",
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