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---
id: 59713da0a428c1a62d7db430
title: 克莱默的统治
title: Cramer's rule
challengeType: 5
videoUrl: ''
forumTopicId: 302239
dashedName: cramers-rule
---
# --description--
<p> <a href='https://en.wikipedia.org/wiki/linear algebra' title='wp：线性代数'>在线性代数中</a> ， <a href='https://en.wikipedia.org/wiki/Cramer&#x27;s rule' title='wp：Cramer的规则'>Cramer规则</a>是一个<a href='https://en.wikipedia.org/wiki/system of linear equations' title='wp：线性方程组'>线性方程组</a>解的显式公式，其中包含与未知数一样多的方程，只要系统具有唯一解，就有效。它通过用方程右边的矢量替换一列来表示（方形）系数矩阵的决定因素和从它获得的矩阵的解决方案。 </p><p>特定</p><p><big></big></p><p> <big>$ \ left \ {\ begin {matrix} a_1x + b_1y + c_1z＆= {\ color {red} d_1} \\ a_2x + b_2y + c_2z＆= {\ color {red} d_2} \\ a_3x + b_3y + c_3z＆= {\颜色{红} D_3} \ {结束矩阵} \权。$</big> </p><p>以矩阵格式表示</p><p><big></big></p><p> <big>$ \ begin {bmatrix} a_1＆b_1＆c_1 \\ a_2＆b_2＆c_2 \\ a_3＆b_3＆c_3 \ end {bmatrix} \ begin {bmatrix} x \\ y \\ z \ end {bmatrix} = \ begin {bmatrix} {\ color {red} d_1} \\ {\ color {red} d_2} \\ {\ color {red} d_3} \ end {bmatrix}。$</big> </p><p>然后可以找到$ x，y $和$ z $的值，如下所示： </p><p><big></big></p><p> <big>$ x = \ frac {\ begin {vmatrix} {\ color {red} d_1}＆b_1＆c_1 \\ {\ color {red} d_2}＆b_2＆c_2 \\ {\ color {red} d_3}＆b_3＆ c_3 \ end {vmatrix}} {\ begin {vmatrix} a_1＆b_1＆c_1 \\ a_2＆b_2＆c_2 \\ a_3＆b_3＆c_3 \ end {vmatrix}}，\ quad y = \ frac {\ begin {vmatrix } a_1＆{\ color {red} d_1}＆c_1 \\ a_2＆{\ color {red} d_2}＆c_2 \\ a_3＆{\ color {red} d_3}＆c_3 \ end {vmatrix}} {\ begin {vmatrix} a_1＆b_1＆c_1 \\ a_2＆b_2＆c_2 \\ a_3＆b_3＆c_3 \ end {vmatrix}}，\ text {和} z = \ frac {\ begin {vmatrix} a_1＆b_1＆{\ color {red} d_1} \\ a_2＆b_2＆{\ color {red} d_2} \\ a_3＆b_3＆{\ color {red} d_3} \ end {vmatrix}} {\ begin {vmatrix} a_1＆b_1＆ c_1 \\ a_2＆b_2＆c_2 \\ a_3＆b_3＆c_3 \ end {vmatrix}}。$</big> </p>任务<p>给定以下方程组： </p><p> <big>$ \ begin {例} 2w-x + 5y + z = -3 \\ 3w + 2x + 2y-6z = -32 \\ w + 3x + 3y-z = -47 \\ 5w-2x-3y + 3z = 49 \\ \ end {cases} $</big> </p><p>使用Cramer的规则解决<big>$ w $，$ x $，$ y $</big>和<big>$ z $</big> 。 </p>
In [linear algebra](https://en.wikipedia.org/wiki/linear algebra "wp: linear algebra"), [Cramer's rule](https://en.wikipedia.org/wiki/Cramer's rule "wp: Cramer's rule") is an explicit formula for the solution of a [system of linear equations](https://en.wikipedia.org/wiki/system of linear equations "wp: system of linear equations") with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.
Given
$\\left\\{\\begin{matrix}a_1x + b_1y + c_1z&= {\\color{red}d_1}\\\\a_2x + b_2y + c_2z&= {\\color{red}d_2}\\\\a_3x + b_3y + c_3z&= {\\color{red}d_3}\\end{matrix}\\right.$
which in matrix format is
$\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix}=\\begin{bmatrix} {\\color{red}d_1} \\\\ {\\color{red}d_2} \\\\ {\\color{red}d_3} \\end{bmatrix}.$
Then the values of $x, y$ and $z$ can be found as follows:
$x = \\frac{\\begin{vmatrix} {\\color{red}d_1} & b_1 & c_1 \\\\ {\\color{red}d_2} & b_2 & c_2 \\\\ {\\color{red}d_3} & b_3 & c_3 \\end{vmatrix} } { \\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix}}, \\quad y = \\frac {\\begin{vmatrix} a_1 & {\\color{red}d_1} & c_1 \\\\ a_2 & {\\color{red}d_2} & c_2 \\\\ a_3 & {\\color{red}d_3} & c_3 \\end{vmatrix}} {\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix}}, \\text{ and }z = \\frac { \\begin{vmatrix} a_1 & b_1 & {\\color{red}d_1} \\\\ a_2 & b_2 & {\\color{red}d_2} \\\\ a_3 & b_3 & {\\color{red}d_3} \\end{vmatrix}} {\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix} }.$
# --instructions--
Given the following system of equations:
$\\begin{cases} 2w-x+5y+z=-3 \\\\ 3w+2x+2y-6z=-32 \\\\ w+3x+3y-z=-47 \\\\ 5w-2x-3y+3z=49 \\\\ \\end{cases}$
solve for $w$, $x$, $y$ and $z$, using Cramer's rule.
# --hints--
`cramersRule`是一个函数。
`cramersRule` should be a function.
```js
assert(typeof cramersRule === 'function');
```
`cramersRule([[2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3]], [-3, -32, -47, 49])`应返回`[2, -12, -4, 1]` 。
`cramersRule([[2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3]], [-3, -32, -47, 49])` should return `[2, -12, -4, 1]`.
```js
assert.deepEqual(cramersRule(matrices[0], freeTerms[0]), answers[0]);
```
`cramersRule([[3, 1, 1], [2, 2, 5], [1, -3, -4]], [3, -1, 2])`应返回`[1, 1, -1]` 。
`cramersRule([[3, 1, 1], [2, 2, 5], [1, -3, -4]], [3, -1, 2])` should return `[1, 1, -1]`.
```js
assert.deepEqual(cramersRule(matrices[1], freeTerms[1]), answers[1]);