{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $K$ be a commutative ring and let $A$ be an $n \\times n$ matrix over $K$.\nThen $A$ is [[invertible|0EB9N0S33C]] if and only if its [[determinant|00KDZRW4C0]] $\\det A$ is a unit of $K$.\nIn particular, over a field, $A$ is invertible if and only if $\\det A \\ne 0$.\n\n## Remarks\n\nIf $\\det A$ is a unit, then the [[adjugate|07W9H8QF38]] gives the inverse $A^{-1} = (\\det A)^{-1} \\operatorname{adj}(A)$.\nOver $\\Z$, the matrix $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$ has determinant $2 \\ne 0$, but it has no inverse with integer entries.\nOver a field, this is one of many equivalent conditions for invertibility, with $\\rank A = n$ and $\\ker A = 0$.\n\n## Explanations\n\n### L1 human\n\nA real square matrix has an inverse exactly when its determinant is not zero.\nA zero determinant means that the matrix squeezes space into a lower dimension.\nSuch a squeeze cannot be undone.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "15A15"
    ],
    "name": "A matrix is invertible iff its determinant is a unit",
    "role": "theorem",
    "sources": [
      {
        "locator": "Invertible matrix theorem",
        "url": "https://en.wikipedia.org/wiki/Invertible_matrix"
      }
    ],
    "tag": "03Q7RE2HXR",
    "uses": [
      "0C3CFHDN6R",
      "0TWTS88TWN",
      "0GMPYM0NSR"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/03Q7RE2HXR/",
  "links": [
    {
      "field": "uses",
      "target": "0C3CFHDN6R"
    },
    {
      "field": "uses",
      "target": "0TWTS88TWN"
    },
    {
      "field": "uses",
      "target": "0GMPYM0NSR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/03Q7RE2HXR.md",
  "msc": [
    "15A15"
  ],
  "name": "A matrix is invertible iff its determinant is a unit",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/03Q/03Q7RE2HXR.md",
  "status": null,
  "tag": "03Q7RE2HXR",
  "uri": "m2://03Q7RE2HXR",
  "url": "https://math2.tamnd.com/wiki/a-matrix-is-invertible-iff-its-determinant-is-a-unit/",
  "writing": {
    "tag": "03Q7RE2HXR",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}