freeCodeCamp

5.7 KiB

Raw Blame History

title	id	challengeType	isHidden	forumTopicId
Cramer's rule	59713da0a428c1a62d7db430	5	false	302239

Description

In linear algebra, Cramer's rule is an explicit formula for the solution of a 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.

Tests

tests:
  - text: <code>cramersRule</code> should be a function.
    testString: assert(typeof cramersRule === 'function');
  - text: <code>cramersRule([[2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3]], [-3, -32, -47, 49])</code> should return <code>[2, -12, -4, 1]</code>.
    testString: assert.deepEqual(cramersRule(matrices[0], freeTerms[0]), answers[0]);
  - text: <code>cramersRule([[3, 1, 1], [2, 2, 5], [1, -3, -4]], [3, -1, 2])</code> should return <code>[1, 1, -1]</code>.
    testString: assert.deepEqual(cramersRule(matrices[1], freeTerms[1]), answers[1]);

Challenge Seed

function cramersRule(matrix, freeTerms) {
  // Good luck!
  return true;
}

After Test

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]];

Solution

/**
 * 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());
}

5.7 KiB Raw Blame History