{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $K$ be a commutative ring and let $A = (a_{ij})$ be an $n \\times n$ upper or lower [[triangular matrix|0N4XY7Z3NN]] over $K$.\nThen\n$$\\det A = a_{11} a_{22} \\cdots a_{nn},$$\nwhere $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nIn the [[Leibniz formula|016PE4VZ8Q]], all terms except the term of the identity permutation contain an entry below, or above, the diagonal, so they are zero.\nRow reduction to a triangular matrix, with this rule, is a practical way to compute determinants.\nA triangular matrix over a field is invertible if and only if all its diagonal entries are nonzero.\n\n## Explanations\n\n### L1 human\n\nFor a triangular matrix, the determinant is the product of the diagonal entries.\nThe entries above the diagonal do not matter.\nFor example, $\\det \\begin{pmatrix} 2 \u0026 7 \\\\ 0 \u0026 3 \\end{pmatrix} = 6$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "15A15"
    ],
    "name": "Determinant of a triangular matrix",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Triangular_matrix"
      }
    ],
    "tag": "0K1KKMFHYH",
    "uses": [
      "0C3CFHDN6R",
      "0MKABWDRNC"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0K1KKMFHYH/",
  "links": [
    {
      "field": "uses",
      "target": "0C3CFHDN6R"
    },
    {
      "field": "uses",
      "target": "0MKABWDRNC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0K1KKMFHYH.md",
  "msc": [
    "15A15"
  ],
  "name": "Determinant of a triangular matrix",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0K1/0K1KKMFHYH.md",
  "status": null,
  "tag": "0K1KKMFHYH",
  "uri": "m2://0K1KKMFHYH",
  "url": "https://math2.tamnd.com/wiki/determinant-of-a-triangular-matrix/",
  "writing": {
    "tag": "0K1KKMFHYH",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}