{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe solve the system with the inverse matrix and read off each coordinate by a Laplace expansion.\n\n1. Let $A = (a_{kl})$ be an $n \\times n$ matrix over the [[field|0QA8CHH8MG]] $F$ with $\\det A \\ne 0$, and let $b = (b_1, \\dots, b_n) \\in F^n$.\n2. First let $n = 1$. Then $A = (a_{11})$ with $a_{11} \\ne 0$, by the [[Leibniz formula|016PE4VZ8Q]] with one term. The equation $a_{11} x_1 = b_1$ has the single solution $x_1 = b_1 / a_{11}$.\n3. For $n = 1$, $A_1 = (b_1)$ and $\\det A_1 = b_1$. So $x_1 = \\det A_1 / \\det A$, as claimed. From now on let $n \\ge 2$.\n4. In a field, the nonzero element $\\det A$ is a unit. By the [[inverse in terms of the adjugate|0ENVM493VK]], $A$ is invertible with $A^{-1} = (\\det A)^{-1} \\operatorname{adj}(A)$.\n5. Existence: let $x = A^{-1} b$. By [[associativity|073E4XC016]], $A x = (A A^{-1}) b = I_n b = b$. So $x$ is a solution of the [[linear system|0JTRMBQ1SA]] $Ax = b$.\n6. Uniqueness: if $A y = b$, then $y = (A^{-1} A) y = A^{-1} (A y) = A^{-1} b = x$, by associativity.\n7. Let $C_{kl}$ be the [[cofactors|0W2QAVJQ1W]] of $A$. By the definition of the [[adjugate|0GMPYM0NSR]] and the [[matrix product|076JGP6R3T]], $x_i = (\\det A)^{-1} \\sum_{j=1}^{n} C_{ji} b_j$.\n8. Fix $i$. The matrix $A_i$ differs from $A$ only in column $i$, and it has entry $b_j$ in position $(j, i)$.\n9. A cofactor in position $(j, i)$ deletes column $i$. So the cofactors of $A_i$ in the positions $(j, i)$ equal the cofactors $C_{ji}$ of $A$.\n10. By the [[Laplace expansion along column $i$|06AMCJMMVM]] of $A_i$ and steps 8 and 9, $\\det A_i = \\sum_{j=1}^{n} b_j C_{ji}$.\n11. By steps 7 and 10, $x_i = \\det A_i / \\det A$ for $1 \\le i \\le n$.\n\n## Idea\n\nThe solution is $x = A^{-1} b = (\\det A)^{-1} \\operatorname{adj}(A)\\, b$. Coordinate $i$ of $\\operatorname{adj}(A)\\, b$ pairs $b$ with the cofactors of column $i$. That pairing is exactly the Laplace expansion of $\\det A_i$ along its new column $i$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "15A06",
      "15A15"
    ],
    "name": "Proof of Cramer's rule by the adjugate",
    "proves": "0977KZ6PQB",
    "sources": [
      {
        "locator": "Proof",
        "url": "https://en.wikipedia.org/wiki/Cramer's_rule"
      }
    ],
    "tag": "4GY0RA4W28",
    "uses": [
      "0ENVM493VK",
      "0GMPYM0NSR",
      "06AMCJMMVM",
      "0W2QAVJQ1W",
      "0TWTS88TWN",
      "073E4XC016",
      "076JGP6R3T",
      "016PE4VZ8Q",
      "0QA8CHH8MG",
      "0JTRMBQ1SA"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/4GY0RA4W28/",
  "links": [
    {
      "field": "proves",
      "target": "0977KZ6PQB"
    },
    {
      "field": "uses",
      "target": "0ENVM493VK"
    },
    {
      "field": "uses",
      "target": "0GMPYM0NSR"
    },
    {
      "field": "uses",
      "target": "06AMCJMMVM"
    },
    {
      "field": "uses",
      "target": "0W2QAVJQ1W"
    },
    {
      "field": "uses",
      "target": "0TWTS88TWN"
    },
    {
      "field": "uses",
      "target": "073E4XC016"
    },
    {
      "field": "uses",
      "target": "076JGP6R3T"
    },
    {
      "field": "uses",
      "target": "016PE4VZ8Q"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "0JTRMBQ1SA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/4GY0RA4W28.md",
  "msc": [
    "15A06",
    "15A15"
  ],
  "name": "Proof of Cramer's rule by the adjugate",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4GY/4GY0RA4W28.md",
  "status": null,
  "tag": "4GY0RA4W28",
  "uri": "m2://4GY0RA4W28",
  "url": "https://math2.tamnd.com/wiki/proof-of-cramers-rule-by-the-adjugate/",
  "writing": {
    "tag": "4GY0RA4W28",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}