在线性代数中 ， Cramer规则是一个线性方程组解的显式公式，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。
特定
$ \ 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。$
以矩阵格式表示
$ \ 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}。$
然后可以找到$ x，y $和$ z $的值，如下所示：
$ 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}}。$
任务给定以下方程组：
$ \ begin {例} 2w-x + 5y + z = -3 \\ 3w + 2x + 2y-6z = -32 \\ w + 3x + 3y-z = -47 \\ 5w-2x-3y + 3z = 49 \\ \ end {cases} $
使用Cramer的规则解决$ w $，$ x $，$ y $和$ z $ 。
# --hints-- `cramersRule`是一个函数。 ```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]` 。 ```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]` 。 ```js assert.deepEqual(cramersRule(matrices[1], freeTerms[1]), answers[1]); ``` # --seed-- ## --after-user-code-- ```js const matrices = [ [ [2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3] ], [ [3, 1, 1], [2, 2, 5], [1, -3, -4] ] ]; const freeTerms = [[-3, -32, -47, 49], [3, -1, 2]]; const answers = [[2, -12, -4, 1], [1, 1, -1]]; ``` ## --seed-contents-- ```js function cramersRule(matrix, freeTerms) { return true; } ``` # --solutions-- ```js /** * Compute Cramer's Rule * @param {array} matrix x,y,z, etc. terms * @param {array} freeTerms * @return {array} solution for x,y,z, etc. */ function cramersRule(matrix, freeTerms) { const det = detr(matrix); const returnArray = []; let i; for (i = 0; i < matrix[0].length; i++) { const tmpMatrix = insertInTerms(matrix, freeTerms, i); returnArray.push(detr(tmpMatrix) / det); } return returnArray; } /** * Inserts single dimensional array into * @param {array} matrix multidimensional array to have ins inserted into * @param {array} ins single dimensional array to be inserted vertically into matrix * @param {array} at zero based offset for ins to be inserted into matrix * @return {array} New multidimensional array with ins replacing the at column in matrix */ function insertInTerms(matrix, ins, at) { const tmpMatrix = clone(matrix); let i; for (i = 0; i < matrix.length; i++) { tmpMatrix[i][at] = ins[i]; } return tmpMatrix; } /** * Compute the determinate of a matrix. No protection, assumes square matrix * function borrowed, and adapted from MIT Licensed numericjs library (www.numericjs.com) * @param {array} m Input Matrix (multidimensional array) * @return {number} result rounded to 2 decimal */ function detr(m) { let ret = 1; let j; let k; const A = clone(m); const n = m[0].length; let alpha; for (j = 0; j < n - 1; j++) { k = j; for (let i = j + 1; i < n; i++) { if (Math.abs(A[i][j]) > Math.abs(A[k][j])) { k = i; } } if (k !== j) { const temp = A[k]; A[k] = A[j]; A[j] = temp; ret *= -1; } const Aj = A[j]; for (let i = j + 1; i < n; i++) { const Ai = A[i]; alpha = Ai[j] / Aj[j]; for (k = j + 1; k < n - 1; k += 2) { const k1 = k + 1; Ai[k] -= Aj[k] * alpha; Ai[k1] -= Aj[k1] * alpha; } if (k !== n) { Ai[k] -= Aj[k] * alpha; } } if (Aj[j] === 0) { return 0; } ret *= Aj[j]; } return Math.round(ret * A[j][j] * 100) / 100; } /** * Clone two dimensional Array using ECMAScript 5 map function and EcmaScript 3 slice * @param {array} m Input matrix (multidimensional array) to clone * @return {array} New matrix copy */ function clone(m) { return m.map(a => a.slice()); } ```