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# Cramer's rule

Cramer's rule
You are encouraged to solve this task according to the task description, using any language you may know.
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

$\left\{{\begin{matrix}a_{1}x+b_{1}y+c_{1}z&={\color {red}d_{1}}\\a_{2}x+b_{2}y+c_{2}z&={\color {red}d_{2}}\\a_{3}x+b_{3}y+c_{3}z&={\color {red}d_{3}}\end{matrix}}\right.$

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:

$x={\frac {\begin{vmatrix}{\color {red}d_{1}}&b_{1}&c_{1}\\{\color {red}d_{2}}&b_{2}&c_{2}\\{\color {red}d_{3}}&b_{3}&c_{3}\end{vmatrix}}{\begin{vmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{vmatrix}}},\quad y={\frac {\begin{vmatrix}a_{1}&{\color {red}d_{1}}&c_{1}\\a_{2}&{\color {red}d_{2}}&c_{2}\\a_{3}&{\color {red}d_{3}}&c_{3}\end{vmatrix}}{\begin{vmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{vmatrix}}},{\text{ and }}z={\frac {\begin{vmatrix}a_{1}&b_{1}&{\color {red}d_{1}}\\a_{2}&b_{2}&{\color {red}d_{2}}\\a_{3}&b_{3}&{\color {red}d_{3}}\end{vmatrix}}{\begin{vmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{vmatrix}}}.$

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.

with Ada.Text_IO;with Ada.Numerics.Generic_Real_Arrays; procedure Cramers_Rules is    type Real is new Float;   --  This is the type we want to use in the matrix and vector    package Real_Arrays is      new Ada.Numerics.Generic_Real_Arrays (Real);    use Real_Arrays;    function Solve_Cramer (M : in Real_Matrix;                          V : in Real_Vector)                         return Real_Vector   is      Denominator : Real;      Nom_Matrix  : Real_Matrix (M'Range (1),                                 M'Range (2));      Numerator   : Real;      Result      : Real_Vector (M'Range (1));   begin      if        M'Length (2) /= V'Length or        M'Length (1) /= M'Length (2)      then         raise Constraint_Error with "Dimensions does not match";      end if;       Denominator := Determinant (M);       for Col in V'Range loop         Nom_Matrix := M;          --  Substitute column         for Row in V'Range loop            Nom_Matrix (Row, Col) := V (Row);         end loop;          Numerator    := Determinant (Nom_Matrix);         Result (Col) := Numerator / Denominator;      end loop;       return Result;   end Solve_Cramer;    procedure Put (V : Real_Vector) is      use Ada.Text_IO;      package Real_IO is         new Ada.Text_IO.Float_IO (Real);   begin      Put ("[");      for E of V loop         Real_IO.Put (E, Exp => 0, Aft => 2);         Put (" ");      end loop;      Put ("]");      New_Line;   end Put;    M : constant Real_Matrix := ((2.0, -1.0,  5.0,  1.0),                                (3.0,  2.0,  2.0, -6.0),                                (1.0,  3.0,  3.0, -1.0),                                (5.0, -2.0, -3.0,  3.0));   V : constant Real_Vector := (-3.0, -32.0, -47.0, 49.0);   R : constant Real_Vector := Solve_Cramer (M, V);begin   Put (R);end Cramers_Rules;
Output:
[ 2.00 -12.00 -4.00  1.00 ]

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses the non-standard DET operator available in Algol 68G.

# returns the solution of a.x = b via Cramer's rule                    ##         this is for REAL arrays, could define additional operators   ##         for INT, COMPL, etc.                                         #PRIO CRAMER = 1;OP   CRAMER = ( [,]REAL a, []REAL b )[]REAL:     IF 1 UPB a /= 2 UPB a     OR 1 LWB a /= 2 LWB a     OR 1 UPB a /=   UPB b     THEN        # the array sizes and bounds do not match                       #        print( ( "Invaid parameters to CRAMER", newline ) );        stop     ELIF REAL deta = DET a;          det a = 0     THEN        # a is singular                                                 #        print( ( "Singular matrix for CRAMER", newline ) );        stop     ELSE        # the arrays have matching bounds                               #        [ LWB b : UPB b ]REAL result;        FOR col FROM LWB b TO UPB b DO            # form a matrix from a with the col'th column replaced by b #            [ 1 LWB a : 1 UPB a, 2 LWB a : 2 UPB a ]REAL m := a;            m[ : , col ] := b[ : AT 1 ];            # col'th result elemet as per Cramer's rule                 #            result[ col ] := DET m / det a        OD;        result     FI; # CRAMER # # test CRAMER using the matrix and column vector specified in the task  #[,]REAL a = ( (  2, -1,  5,  1 )            , (  3,  2,  2, -6 )            , (  1,  3,  3, -1 )            , (  5, -2, -3,  3 )            );[]REAL  b = (  -3            , -32            , -47            ,  49            );[]REAL  solution = a CRAMER b;FOR c FROM LWB solution TO UPB solution DO    print( ( " ", fixed( solution[ c ], -8, 4 ) ) )OD;print( ( newline ) )
Output:
   2.0000 -12.0000  -4.0000   1.0000


## C

#include <math.h>#include <stdio.h>#include <stdlib.h> typedef struct {    int n;    double **elems;} SquareMatrix; SquareMatrix init_square_matrix(int n, double elems[n][n]) {    SquareMatrix A = {        .n = n,        .elems = malloc(n * sizeof(double *))    };    for(int i = 0; i < n; ++i) {        A.elems[i] = malloc(n * sizeof(double));        for(int j = 0; j < n; ++j)            A.elems[i][j] = elems[i][j];    }     return A;} SquareMatrix copy_square_matrix(SquareMatrix src) {    SquareMatrix dest;    dest.n = src.n;    dest.elems = malloc(dest.n * sizeof(double *));    for(int i = 0; i < dest.n; ++i) {        dest.elems[i] = malloc(dest.n * sizeof(double));        for(int j = 0; j < dest.n; ++j)            dest.elems[i][j] = src.elems[i][j];    }     return dest;} double det(SquareMatrix A) {    double det = 1;     for(int j = 0; j < A.n; ++j) {        int i_max = j;        for(int i = j; i < A.n; ++i)            if(A.elems[i][j] > A.elems[i_max][j])                i_max = i;         if(i_max != j) {            for(int k = 0; k < A.n; ++k) {                double tmp = A.elems[i_max][k];                A.elems[i_max][k] = A.elems[j][k];                A.elems[j][k]     = tmp;            }             det *= -1;        }         if(abs(A.elems[j][j]) < 1e-12) {            puts("Singular matrix!");            return NAN;        }         for(int i = j + 1; i < A.n; ++i) {            double mult = -A.elems[i][j] / A.elems[j][j];            for(int k = 0; k < A.n; ++k)                A.elems[i][k] += mult * A.elems[j][k];        }    }     for(int i = 0; i < A.n; ++i)        det *= A.elems[i][i];     return det;} void deinit_square_matrix(SquareMatrix A) {    for(int i = 0; i < A.n; ++i)        free(A.elems[i]);    free(A.elems);} double cramer_solve(SquareMatrix A, double det_A, double *b, int var) {    SquareMatrix tmp = copy_square_matrix(A);    for(int i = 0; i < tmp.n; ++i)        tmp.elems[i][var] = b[i];     double det_tmp = det(tmp);    deinit_square_matrix(tmp);     return det_tmp / det_A;} int main(int argc, char **argv) {#define N 4    double elems[N][N] = {        { 2, -1,  5,  1},        { 3,  2,  2, -6},        { 1,  3,  3, -1},        { 5, -2, -3,  3}    };    SquareMatrix A = init_square_matrix(N, elems);     SquareMatrix tmp = copy_square_matrix(A);    int det_A = det(tmp);    deinit_square_matrix(tmp);     double b[] = {-3, -32, -47, 49};     for(int i = 0; i < N; ++i)        printf("%7.3lf\n", cramer_solve(A, det_A, b, i));     deinit_square_matrix(A);    return EXIT_SUCCESS;}
Output:
  2.000
-12.000
-4.000
1.000

## C#

Instead of copying a bunch of arrays, I made a class with an indexer that simply accesses the correct items in the original matrix.

using System;using System.Collections.Generic;using static System.Linq.Enumerable; public static class CramersRule{    public static void Main() {        var equations = new [] {            new [] { 2, -1,  5,  1,  -3 },            new [] { 3,  2,  2, -6, -32 },            new [] { 1,  3,  3, -1, -47 },            new [] { 5, -2, -3,  3,  49 }        };        var solution = SolveCramer(equations);        Console.WriteLine(solution.DelimitWith(", "));    }     public static int[] SolveCramer(int[][] equations) {        int size = equations.Length;        if (equations.Any(eq => eq.Length != size + 1)) throw new ArgumentException($"Each equation must have {size+1} terms."); int[,] matrix = new int[size, size]; int[] column = new int[size]; for (int r = 0; r < size; r++) { column[r] = equations[r][size]; for (int c = 0; c < size; c++) { matrix[r, c] = equations[r][c]; } } return Solve(new SubMatrix(matrix, column)); } private static int[] Solve(SubMatrix matrix) { int det = matrix.Det(); if (det == 0) throw new ArgumentException("The determinant is zero."); int[] answer = new int[matrix.Size]; for (int i = 0; i < matrix.Size; i++) { matrix.ColumnIndex = i; answer[i] = matrix.Det() / det; } return answer; } //Extension method from library. static string DelimitWith<T>(this IEnumerable<T> source, string separator = " ") => string.Join(separator ?? " ", source ?? Empty<T>()); private class SubMatrix { private int[,] source; private SubMatrix prev; private int[] replaceColumn; public SubMatrix(int[,] source, int[] replaceColumn) { this.source = source; this.replaceColumn = replaceColumn; this.prev = null; this.ColumnIndex = -1; Size = replaceColumn.Length; } private SubMatrix(SubMatrix prev, int deletedColumnIndex = -1) { this.source = null; this.prev = prev; this.ColumnIndex = deletedColumnIndex; Size = prev.Size - 1; } public int ColumnIndex { get; set; } public int Size { get; } public int this[int row, int column] { get { if (source != null) return column == ColumnIndex ? replaceColumn[row] : source[row, column]; return prev[row + 1, column < ColumnIndex ? column : column + 1]; } } public int Det() { if (Size == 1) return this[0, 0]; if (Size == 2) return this[0, 0] * this[1, 1] - this[0, 1] * this[1, 0]; SubMatrix m = new SubMatrix(this); int det = 0; int sign = 1; for (int c = 0; c < Size; c++) { m.ColumnIndex = c; int d = m.Det(); det += this[0, c] * d * sign; sign = -sign; } return det; } public void Print() { for (int r = 0; r < Size; r++) { Console.WriteLine(Range(0, Size).Select(c => this[r, c]).DelimitWith(", ")); } Console.WriteLine(); } } } Output: 2, -12, -4, 1  ## C++ Translation of: C# #include <algorithm>#include <iostream>#include <vector> class SubMatrix { const std::vector<std::vector<double>> *source; std::vector<double> replaceColumn; const SubMatrix *prev; size_t sz; int colIndex = -1; public: SubMatrix(const std::vector<std::vector<double>> &src, const std::vector<double> &rc) : source(&src), replaceColumn(rc), prev(nullptr), colIndex(-1) { sz = replaceColumn.size(); } SubMatrix(const SubMatrix &p) : source(nullptr), prev(&p), colIndex(-1) { sz = p.size() - 1; } SubMatrix(const SubMatrix &p, int deletedColumnIndex) : source(nullptr), prev(&p), colIndex(deletedColumnIndex) { sz = p.size() - 1; } int columnIndex() const { return colIndex; } void columnIndex(int index) { colIndex = index; } size_t size() const { return sz; } double index(int row, int col) const { if (source != nullptr) { if (col == colIndex) { return replaceColumn[row]; } else { return (*source)[row][col]; } } else { if (col < colIndex) { return prev->index(row + 1, col); } else { return prev->index(row + 1, col + 1); } } } double det() const { if (sz == 1) { return index(0, 0); } if (sz == 2) { return index(0, 0) * index(1, 1) - index(0, 1) * index(1, 0); } SubMatrix m(*this); double det = 0.0; int sign = 1; for (size_t c = 0; c < sz; ++c) { m.columnIndex(c); double d = m.det(); det += index(0, c) * d * sign; sign = -sign; } return det; }}; std::vector<double> solve(SubMatrix &matrix) { double det = matrix.det(); if (det == 0.0) { throw new std::runtime_error("The determinant is zero."); } std::vector<double> answer(matrix.size()); for (int i = 0; i < matrix.size(); ++i) { matrix.columnIndex(i); answer[i] = matrix.det() / det; } return answer;} std::vector<double> solveCramer(const std::vector<std::vector<double>> &equations) { int size = equations.size(); if (std::any_of( equations.cbegin(), equations.cend(), [size](const std::vector<double> &a) { return a.size() != size + 1; } )) { throw new std::runtime_error("Each equation must have the expected size."); } std::vector<std::vector<double>> matrix(size); std::vector<double> column(size); for (int r = 0; r < size; ++r) { column[r] = equations[r][size]; matrix[r].resize(size); for (int c = 0; c < size; ++c) { matrix[r][c] = equations[r][c]; } } SubMatrix sm(matrix, column); return solve(sm);} template<typename T>std::ostream &operator<<(std::ostream &os, const std::vector<T> &v) { auto it = v.cbegin(); auto end = v.cend(); os << '['; if (it != end) { os << *it++; } while (it != end) { os << ", " << *it++; } return os << ']';} int main() { std::vector<std::vector<double>> equations = { { 2, -1, 5, 1, -3}, { 3, 2, 2, -6, -32}, { 1, 3, 3, -1, -47}, { 5, -2, -3, 3, 49}, }; auto solution = solveCramer(equations); std::cout << solution << '\n'; return 0;} Output: [2, -12, -4, 1] ## Common Lisp (defun minor (m col) (loop with dim = (1- (array-dimension m 0)) with result = (make-array (list dim dim)) for i from 1 to dim for r = (1- i) do (loop with c = 0 for j to dim when (/= j col) do (setf (aref result r c) (aref m i j)) (incf c)) finally (return result))) (defun det (m) (assert (= (array-rank m) 2)) (assert (= (array-dimension m 0) (array-dimension m 1))) (let ((dim (array-dimension m 0))) (if (= dim 1) (aref m 0 0) (loop for col below dim for sign = 1 then (- sign) sum (* sign (aref m 0 col) (det (minor m col))))))) (defun replace-column (m col values) (let* ((dim (array-dimension m 0)) (result (make-array (list dim dim)))) (dotimes (r dim result) (dotimes (c dim) (setf (aref result r c) (if (= c col) (aref values r) (aref m r c))))))) (defun solve (m v) (loop with dim = (array-dimension m 0) with det = (det m) for col below dim collect (/ (det (replace-column m col v)) det))) (solve #2A((2 -1 5 1) (3 2 2 -6) (1 3 3 -1) (5 -2 -3 3)) #(-3 -32 -47 49)) Output: (2 -12 -4 1) ## D Translation of: Kotlin import std.array : array, uninitializedArray;import std.range : iota;import std.stdio : writeln;import std.typecons : tuple; alias vector = double;alias matrix = vector; auto johnsonTrotter(int n) { auto p = iota(n).array; auto q = iota(n).array; auto d = uninitializedArray!(int[])(n); d[] = -1; auto sign = 1; int[][] perms; int[] signs; void permute(int k) { if (k >= n) { perms ~= p.dup; signs ~= sign; sign *= -1; return; } permute(k + 1); foreach (i; 0..k) { auto z = p[q[k] + d[k]]; p[q[k]] = z; p[q[k] + d[k]] = k; q[z] = q[k]; q[k] += d[k]; permute(k + 1); } d[k] *= -1; } permute(0); return tuple!("sigmas", "signs")(perms, signs);} auto determinant(matrix m) { auto jt = johnsonTrotter(m.length); auto sum = 0.0; foreach (i,sigma; jt.sigmas) { auto prod = 1.0; foreach (j,s; sigma) { prod *= m[j][s]; } sum += jt.signs[i] * prod; } return sum;} auto cramer(matrix m, vector d) { auto divisor = determinant(m); auto numerators = uninitializedArray!(matrix[])(m.length); foreach (i; 0..m.length) { foreach (j; 0..m.length) { foreach (k; 0..m.length) { numerators[i][j][k] = m[j][k]; } } } vector v; foreach (i; 0..m.length) { foreach (j; 0..m.length) { numerators[i][j][i] = d[j]; } } foreach (i; 0..m.length) { v[i] = determinant(numerators[i]) / divisor; } return v;} void main() { matrix m = [ [2.0, -1.0, 5.0, 1.0], [3.0, 2.0, 2.0, -6.0], [1.0, 3.0, 3.0, -1.0], [5.0, -2.0, -3.0, 3.0] ]; vector d = [-3.0, -32.0, -47.0, 49.0]; auto wxyz = cramer(m, d); writeln("w = ", wxyz, ", x = ", wxyz, ", y = ", wxyz, ", z = ", wxyz);} Output: w = 2, x = -12, y = -4, z = 1 ## EchoLisp  (lib 'matrix)(string-delimiter "")(define (cramer A B (X)) ;; --> vector X (define ∆ (determinant A)) (for/vector [(i (matrix-col-num A))] (set! X (matrix-set-col! (array-copy A) i B)) (// (determinant X) ∆))) (define (task) (define A (list->array '( 2 -1 5 1 3 2 2 -6 1 3 3 -1 5 -2 -3 3) 4 4)) (define B #(-3 -32 -47 49)) (writeln "Solving A * X = B") (array-print A) (writeln "B = " B) (writeln "X = " (cramer A B)))  Output: (task) Solving A * X = B 2 -1 5 1 3 2 2 -6 1 3 3 -1 5 -2 -3 3 B = #( -3 -32 -47 49) X = #( 2 -12 -4 1)  ## Factor USING: kernel math math.matrices.laplace prettyprint sequences ;IN: rosetta-code.cramers-rule : replace-col ( elt n seq -- seq' ) flip [ set-nth ] keep flip ; : solve ( m v -- seq ) dup length <iota> [ rot [ replace-col ] keep [ determinant ] [email protected] / ] 2with map ; : cramers-rule-demo ( -- ) { { 2 -1 5 1 } { 3 2 2 -6 } { 1 3 3 -1 } { 5 -2 -3 3 } } { -3 -32 -47 49 } solve . ; MAIN: cramers-rule-demo Output: { 2 -12 -4 1 }  ## Fortran In Numerical Methods That Work (Usually), in the section What not to compute, F. S. Acton remarks "...perhaps we should be glad he didn't resort to Cramer's rule (still taught as the practical method in some high schools) and solve his equations as the ratios of determinants - a process that requires labor proportional to N! if done in the schoolboy manner. The contrast with N3 can be startling!" And further on, "Having hinted darkly at my computational fundamentalism, it is probably time to commit to a public heresy by denouncing recursive calculations. I have never seen a numerical problem arising from the physical world that was best calculated by a recursive subroutine..." Since this problem requires use of Cramer's rule, one might as well be hung for a sheep instead of a lamb, so the traditions of Old Fortran and heavy computation will be ignored and the fearsome RECURSIVE specification employed so that the determinants will be calculated recursively, all the way down to N = 1 even though the N = 2 case is easy. This requires F90 and later. Similarly, the MODULE protocol will be employed, even though there is no significant context to share. The alternative method for calculating a determinant involves generating permutations, a tiresome process. Array passing via the modern arrangements of F90 is a source of novel difficulty to set against the slight convenience of not having to pass an additional parameter, N. Explicitly, at least. There are "secret" additional parameters when an array is being passed in the modern way, which are referred to by the new SIZE function. Anyway, for an order N square matrix, the array must be declared A(N,N), and specifically not something like A(100,100) with usage only of elements up to N = 7, say, because the locations in storage of elements in use would be quite different from those used by an array declared A(7,7). This means that the array must be re-declared for each different size usage, a tiresome and error-inviting task. One-dimensional arrays do not have this problem, but they do have to be "long enough" so B and X might as well be included. This also means that the auxiliary matrices needed within the routines have to be made the right size, and fortunately they can be declared in a way that requests this without the blather of ALLOCATE, this being a protocol introduced by Algol in the 1960s. Unfortunately, there is no scheme such as in pl/i to declare AUX "like" A, so some grotesquery results. And in the case of function DET which needs an array of order N - 1, when its recursion bottoms out with N = 1 it will have declared MINOR(0,0), a rather odd situation that fortunately evokes no complaint, and a test run in which its "value" was written out by WRITE (6,*) MINOR produced a blank line: no complaint there either, presumably because zero elements were being sent forth and so there was no improper access of ... nothing. With matrices, there is a problem all the way from the start in 1958. Everyone agrees that a matrix should be indexed as A(row,column) and that when written out, rows should run down the page and columns across. This is unexceptional and the F90 function MATMUL follows this usage. However, Fortran has always stored its array elements in what is called "column major" order, which is to say that element A(1,1) is followed by element A(2,1) in storage, not A(1,2). Thus, if an array is written (or read) by something like WRITE (6,*) A, consecutive elements, written along a line, will be A(1,1), A(2,1), A(3,1), ... So, subroutine SHOWMATRIX is employed to write the matrix out in the desired form, and to read the values into the array, an explicit loop is used to place them where expected rather than just READ(INF,*) A Similarly, if instead a DATA statement were used to initialise the array for the example problem, and it looked something like  DATA A/2, -1, 5, 1 1 3, 2, 2, -6 2 1, 3, 3, -1 3 5, -2, -3, 3/ (ignoring integer/floating-point issues) thus corresponding to the layout of the example problem, there would need to be a statement A = TRANSPOSE(A) to obtain the required order. I have never seen an explanation of why this choice was made for Fortran.  MODULE BAD IDEA !Employ Cramer's rule for solving A.x = b... INTEGER MSG !Might as well be here. CONTAINS !The details. SUBROUTINE SHOWMATRIX(A) !With nice vertical bars. DOUBLE PRECISION A(:,:) !The matrix. INTEGER R,N !Assistants. N = SIZE(A, DIM = 1) !Instead of passing an explicit parameter. DO R = 1,N !Work down the rows. WRITE (MSG,1) A(R,:) !Fling forth a row at a go. 1 FORMAT ("|",<N>F12.3,"|") !Bounded by bars. END DO !On to the next row. END SUBROUTINE SHOWMATRIX !Furrytran's default order is the transpose. RECURSIVE DOUBLE PRECISION FUNCTION DET(A) !Determine the determinant. DOUBLE PRECISION A(:,:) !The square matrix, order N. DOUBLE PRECISION MINOR(SIZE(A,DIM=1) - 1,SIZE(A,DIM=1) - 1) !Order N - 1. DOUBLE PRECISION D !A waystation. INTEGER C,N !Steppers. N = SIZE(A, DIM = 1) !Suplied via secret parameters. IF (N .LE. 0) THEN !This really ought not happen. STOP "DET: null array!" !But I'm endlessly suspicious. ELSE IF (N .NE. SIZE(A, DIM = 2)) THEN !And I'd rather have a decent message STOP "DET: not a square array!" !In place of a crashed run. ELSE IF (N .EQ. 1) THEN !Alright, now get on with it. DET = A(1,1) !This is really easy. ELSE !But otherwise, D = 0 !Here we go. DO C = 1,N !Step along the columns of the first row. CALL FILLMINOR(C) !Produce the auxiliary array for each column. IF (MOD(C,2) .EQ. 0) THEN !Odd or even case? D = D - A(1,C)*DET(MINOR) !Even: subtract. ELSE !Otherwise, D = D + A(1,C)*DET(MINOR) !Odd: add. END IF !So much for that term. END DO !On to the next. DET = D !Declare the result. END IF !So much for that. CONTAINS !An assistant. SUBROUTINE FILLMINOR(CC) !Corresponding to A(1,CC). INTEGER CC !The column being omitted. INTEGER R !A stepper. DO R = 2,N !Ignoring the first row, MINOR(R - 1,1:CC - 1) = A(R,1:CC - 1) !Copy columns 1 to CC - 1. There may be none. MINOR(R - 1,CC:) = A(R,CC + 1:) !And from CC + 1 to N. There may be none. END DO !On to the next row. END SUBROUTINE FILLMINOR !Divide and conquer. END FUNCTION DET !Rather than mess with permutations. SUBROUTINE CRAMER(A,X,B) !Solve A.x = b, where A is a matrix...Careful! The array must be A(N,N), and not say A(100,100) of which only up to N = 6 are in use. DOUBLE PRECISION A(:,:) !A square matrix. I hope. DOUBLE PRECISION X(:),B(:) !Column matrices look rather like 1-D arrays. DOUBLE PRECISION AUX(SIZE(A,DIM=1),SIZE(A,DIM=1)) !Can't say "LIKE A", as in pl/i, alas. DOUBLE PRECISION D !To be calculated once. INTEGER N !The order of the square matrix. I hope. INTEGER C !A stepper. N = SIZE(A, DIM = 1) !Alright, what's the order of battle? D = DET(A) !Once only. IF (D.EQ.0) STOP "Cramer: zero determinant!" !Surely, this won't happen... AUX = A !Prepare the assistant. DO C = 1,N !Step across the columns. IF (C.GT.1) AUX(1:N,C - 1) = A(1:N,C - 1) !Repair previous damage. AUX(1:N,C) = B(1:N) !Place the current damage. X(C) = DET(AUX)/D !The result! END DO !On to the next column. END SUBROUTINE CRAMER !This looks really easy! END MODULE BAD IDEA !But actually, it is a bad idea for N > 2. PROGRAM TEST !Try it and see. USE BAD IDEA !Just so. DOUBLE PRECISION, ALLOCATABLE ::A(:,:), B(:), X(:) !Custom work areas. INTEGER N,R !Assistants.. INTEGER INF !An I/O unit. MSG = 6 !Output. INF = 10 !For reading test data. OPEN (INF,FILE="Test.dat",STATUS="OLD",ACTION="READ") !As in this file.. Chew into the next problem. 10 IF (ALLOCATED(A)) DEALLOCATE(A) !First, IF (ALLOCATED(B)) DEALLOCATE(B) !Get rid of IF (ALLOCATED(X)) DEALLOCATE(X) !The hired help. READ (INF,*,END = 100) N !Since there is a new order. IF (N.LE.0) GO TO 100 !Perhaps a final order. WRITE (MSG,11) N !Othewise, announce prior to acting. 11 FORMAT ("Order ",I0," matrix A, as follows...") !In case something goes wrong. ALLOCATE(A(N,N)) !For instance, ALLOCATE(B(N)) !Out of memory. ALLOCATE(X(N)) !But otherwise, a tailored fit. DO R = 1,N !Now read in the values for the matrix. READ(INF,*,END=667,ERR=665) A(R,:),B(R) !One row of A at a go, followed by B's value. END DO !In free format. CALL SHOWMATRIX(A) !Show what we have managed to obtain. WRITE (MSG,12) "In the scheme A.x = b, b = ",B !In case something goes wrong. 12 FORMAT (A,<N>F12.6) !How many would be too many? CALL CRAMER(A,X,B) !The deed! WRITE (MSG,12) " Via Cramer's rule, x = ",X !The result! GO TO 10 !And try for another test problem. Complaints. 665 WRITE (MSG,666) "Format error",R !I know where I came from. 666 FORMAT (A," while reading row ",I0,"!") !So I can refer to R. GO TO 100 !So much for that. 667 WRITE (MSG,666) "End-of-file", R !Some hint as to where. Closedown. 100 WRITE (6,*) "That was interesting." !Quite. END !Open files are closed, allocated memory is released.  Oddly, the Compaq Visual Fortran F90/95 compiler is confused by the usage "BAD IDEA" instead of "BADIDEA" - spaces are not normally relevant in Fortran source. Anyway, file Test.dat has been filled with the numbers of the example, as follows: 4 /The order, for A.x = b. 2 -1 5 1, -3 /First row of A, b 3 2 2 -6, -32 /Second row... 1 3 3 -1, -47 third row. 5 -2 -3 3, 49 /Last row.  Fortran's free-form allows a comma, a tab, and spaces between numbers, and regards the / as starting a comment, but, because each row is read separately, once the required five (N + 1) values have been read, no further scanning of the line takes place and the next READ statement will start with a new line of input. So the / isn't needed for the third row, as shown. Omitted values lead to confusion as the input process would read additional lines to fill the required count and everything gets out of step. Echoing input very soon after it is obtained is helpful in making sense of such mistakes. For more practical use it would probably be better to constrain the freedom somewhat, perhaps requiring that all the N + 1 values for a row appear on one input record. In such a case, the record could first be read into a text variable (from which the data would be read) so that if a problem arises the text could be printed as a part of the error message. But, this requires guessing a suitably large length for the text variable so as to accommodate the longest possible input line. Output: Order 4 matrix A, as follows... | 2.000 -1.000 5.000 1.000| | 3.000 2.000 2.000 -6.000| | 1.000 3.000 3.000 -1.000| | 5.000 -2.000 -3.000 3.000| In the scheme A.x = b, b = -3.000000 -32.000000 -47.000000 49.000000 Via Cramer's rule, x = 2.000000 -12.000000 -4.000000 1.000000 That was interesting.  And at this point I suddenly noticed that the habits of Old Fortran are not so easily suppressed: all calculations are done with double precision. Curiously enough, for the specific example data, the same results are obtained if all variables are integer. ## Go Library gonum: package main import ( "fmt" "gonum.org/v1/gonum/mat") var m = mat.NewDense(4, 4, []float64{ 2, -1, 5, 1, 3, 2, 2, -6, 1, 3, 3, -1, 5, -2, -3, 3,}) var v = []float64{-3, -32, -47, 49} func main() { x := make([]float64, len(v)) b := make([]float64, len(v)) d := mat.Det(m) for c := range v { mat.Col(b, c, m) m.SetCol(c, v) x[c] = mat.Det(m) / d m.SetCol(c, b) } fmt.Println(x)} Output: [2 -12.000000000000007 -4.000000000000001 1.0000000000000009]  Library go.matrix: package main import ( "fmt" "github.com/skelterjohn/go.matrix") var m = matrix.MakeDenseMatrixStacked([][]float64{ {2, -1, 5, 1}, {3, 2, 2, -6}, {1, 3, 3, -1}, {5, -2, -3, 3},}) var v = []float64{-3, -32, -47, 49} func main() { x := make([]float64, len(v)) b := make([]float64, len(v)) d := m.Det() for c := range v { m.BufferCol(c, b) m.FillCol(c, v) x[c] = m.Det() / d m.FillCol(c, b) } fmt.Println(x)} Output: [2.0000000000000004 -11.999999999999998 -4 0.9999999999999999]  ## Haskell ### Version 1 import Data.Matrix solveCramer :: (Ord a, Fractional a) => Matrix a -> Matrix a -> Maybe [a]solveCramer a y | da == 0 = Nothing | otherwise = Just$ map (\i -> d i / da) [1..n]  where da = detLU a        d i = detLU $submatrix 1 n 1 n$ switchCols i (n+1) ay        ay = a <|> y        n = ncols a task = solveCramer a y  where a = fromLists [[2,-1, 5, 1]                      ,[3, 2, 2,-6]                      ,[1, 3, 3,-1]                      ,[5,-2,-3, 3]]        y = fromLists [[-3], [-32], [-47], ]
Output:
λ> task
Just [2.0000000000000004,-11.999999999999998,-4.0,0.9999999999999999]


### Version 2

We use Rational numbers for having more precision. a % b is the rational a / b.

s_permutations :: [a] -> [([a], Int)]s_permutations = flip zip (cycle [1, -1]) . (foldl aux [[]])  where aux items x = do          (f,item) <- zip (cycle [reverse,id]) items          f (insertEv x item)        insertEv x [] = [[x]]        insertEv x l@(y:ys) = (x:l) :  map (y:) (insertEv x ys) mult:: Num a => [[a]] -> [[a]] -> [[a]]mult uss vss = map ((\xs -> if null xs then [] else foldl1 (zipWith (+)) xs). zipWith (\vs u -> map (u*) vs) vss) uss matI::(Num a) => Int -> [[a]]matI n = [ [fromIntegral.fromEnum $i == j | i <- [1..n]] | j <- [1..n]] elemPos::[[a]] -> Int -> Int -> aelemPos ms i j = (ms !! i) !! j prod:: Num a => ([[a]] -> Int -> Int -> a) -> [[a]] -> [Int] -> aprod f ms = product.zipWith (f ms) [0..] s_determinant:: Num a => ([[a]] -> Int -> Int -> a) -> [[a]] -> [([Int],Int)] -> as_determinant f ms = sum.map (\(is,s) -> fromIntegral s * prod f ms is) elemCramerPos::Int -> Int -> [[a]] -> [[a]] -> Int -> Int -> aelemCramerPos l k ks ms i j = if j /= l then elemPos ms i j else elemPos ks i k solveCramer:: [[Rational]] -> [[Rational]] -> [[Rational]]solveCramer ms ks = xs where xs | d /= 0 = go (reverse [0..pred.length.head$ ks])     | otherwise = []  go (u:us) = foldl glue (col u) us  glue us u = zipWith (\ys (y:_) -> y:ys) us (col u)  col k = map (\l -> [(/d) $s_determinant (elemCramerPos l k ks) ms ps])$ ls  ls = [0..pred.length $ms] ps = s_permutations ls d = s_determinant elemPos ms ps task::[[Rational]] -> [[Rational]] -> IO()task a b = do let x = solveCramer a b let u = map (map fromRational) x let y = mult a x let identity = matI (length x) let a1 = solveCramer a identity let h = mult a a1 let z = mult a1 b putStrLn "a =" mapM_ print a putStrLn "b =" mapM_ print b putStrLn "solve: a * x = b => x = solveCramer a b =" mapM_ print x putStrLn "u = fromRationaltoDouble x =" mapM_ print u putStrLn "verification: y = a * x = mult a x =" mapM_ print y putStrLn$ "test: y == b = "  print $y == b putStrLn "identity matrix: identity =" mapM_ print identity putStrLn "find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveCramer a identity =" mapM_ print a1 putStrLn "verification: h = a * a1 = mult a a1 =" mapM_ print h putStrLn$ "test: h == identity = "  print $h == identity putStrLn "z = a1 * b = mult a1 b =" mapM_ print z putStrLn "test: z == x =" print$ z == x main = do  let a  = [[2,-1, 5, 1]           ,[3, 2, 2,-6]           ,[1, 3, 3,-1]           ,[5,-2,-3, 3]]  let b   =  [[-3], [-32], [-47], ]  task a b
Output:
a =
[2 % 1,(-1) % 1,5 % 1,1 % 1]
[3 % 1,2 % 1,2 % 1,(-6) % 1]
[1 % 1,3 % 1,3 % 1,(-1) % 1]
[5 % 1,(-2) % 1,(-3) % 1,3 % 1]
b =
[(-3) % 1]
[(-32) % 1]
[(-47) % 1]
[49 % 1]
solve: a * x = b => x = solveCramer a b =
[2 % 1]
[(-12) % 1]
[(-4) % 1]
[1 % 1]
u = fromRationaltoDouble x =
[2.0]
[-12.0]
[-4.0]
[1.0]
verification: y = a * x = mult a x =
[(-3) % 1]
[(-32) % 1]
[(-47) % 1]
[49 % 1]
test: y == b =
True
identity matrix: identity =
[1 % 1,0 % 1,0 % 1,0 % 1]
[0 % 1,1 % 1,0 % 1,0 % 1]
[0 % 1,0 % 1,1 % 1,0 % 1]
[0 % 1,0 % 1,0 % 1,1 % 1]
find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveCramer a identity =
[4 % 171,11 % 171,10 % 171,8 % 57]
[(-55) % 342,(-23) % 342,119 % 342,2 % 57]
[107 % 684,(-5) % 684,11 % 684,(-7) % 114]
[7 % 684,(-109) % 684,103 % 684,7 % 114]
verification: h = a * a1 = mult a a1 =
[1 % 1,0 % 1,0 % 1,0 % 1]
[0 % 1,1 % 1,0 % 1,0 % 1]
[0 % 1,0 % 1,1 % 1,0 % 1]
[0 % 1,0 % 1,0 % 1,1 % 1]
test: h == identity =
True
z = a1 * b = mult a1 b =
[2 % 1]
[(-12) % 1]
[(-4) % 1]
[1 % 1]
test: z == x =
True


### Version 3

import Data.List determinant::(Fractional a, Ord a) => [[a]] -> adeterminant ls = if null ls then 0 else pivot 1 (zip ls [(0::Int)..])  where  good rs ts = (abs.head.fst $ts) <= (abs.head.fst$ rs)  go us (vs,i) = if v == 0 then (ws,i) else (zipWith (\x y -> y - x*v) us ws,i)    where (v,ws) = (head $vs,tail vs) change i (ys:zs) = map (\xs -> if (==i).snd$ xs then ys else xs) zs  pivot d [] = d  pivot d zs@((_,j):ys) = if 0 == u then 0 else pivot e ws    where    e = if i == j then u*d else -u*d    ((u:us),i) = foldl1 (\rs ts ->  if good rs ts then rs else ts) zs    ws = map (go (map (/u) us)) $if i == j then ys else change i zs solveCramer::(Fractional a, Ord a) => [[a]] -> [[a]] -> [[a]]solveCramer as bs = if 0 == d then [] else ans bs where d = determinant as ans = transpose.map go.transpose where ms = zip [0..] (transpose as) go us = [ (/d)$ determinant [if i /= j then vs else us | (j,vs) <- ms] | (i,_) <- ms] matI::(Num a) => Int -> [[a]]matI n = [ [fromIntegral.fromEnum $i == j | i <- [1..n]] | j <- [1..n]] mult:: Num a => [[a]] -> [[a]] -> [[a]]mult uss vss = map ((\xs -> if null xs then [] else foldl1 (zipWith (+)) xs). zipWith (\vs u -> map (u*) vs) vss) uss task::[[Rational]] -> [[Rational]] -> IO()task a b = do let x = solveCramer a b let u = map (map fromRational) x let y = mult a x let identity = matI (length x) let a1 = solveCramer a identity let h = mult a a1 let z = mult a1 b putStrLn "a =" mapM_ print a putStrLn "b =" mapM_ print b putStrLn "solve: a * x = b => x = solveCramer a b =" mapM_ print x putStrLn "u = fromRationaltoDouble x =" mapM_ print u putStrLn "verification: y = a * x = mult a x =" mapM_ print y putStrLn$ "test: y == b = "  print $y == b putStrLn "identity matrix: identity =" mapM_ print identity putStrLn "find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveCramer a identity =" mapM_ print a1 putStrLn "verification: h = a * a1 = mult a a1 =" mapM_ print h putStrLn$ "test: h == identity = "  print $h == identity putStrLn "z = a1 * b = mult a1 b =" mapM_ print z putStrLn "test: z == x =" print$ z == x main = do  let a  = [[2,-1, 5, 1]           ,[3, 2, 2,-6]           ,[1, 3, 3,-1]           ,[5,-2,-3, 3]]  let b  = [[-3], [-32], [-47], ]  task a b
Output:
a =
[2 % 1,(-1) % 1,5 % 1,1 % 1]
[3 % 1,2 % 1,2 % 1,(-6) % 1]
[1 % 1,3 % 1,3 % 1,(-1) % 1]
[5 % 1,(-2) % 1,(-3) % 1,3 % 1]
b =
[(-3) % 1]
[(-32) % 1]
[(-47) % 1]
[49 % 1]
solve: a * x = b => x = solveCramer a b =
[2 % 1]
[(-12) % 1]
[(-4) % 1]
[1 % 1]
u = fromRationaltoDouble x =
[2.0]
[-12.0]
[-4.0]
[1.0]
verification: y = a * x = mult a x =
[(-3) % 1]
[(-32) % 1]
[(-47) % 1]
[49 % 1]
test: y == b =
True
identity matrix: identity =
[1 % 1,0 % 1,0 % 1,0 % 1]
[0 % 1,1 % 1,0 % 1,0 % 1]
[0 % 1,0 % 1,1 % 1,0 % 1]
[0 % 1,0 % 1,0 % 1,1 % 1]
find: a1 = inv(a) => solve: a * a1 = identity => a1 = solveCramer a identity =
[4 % 171,11 % 171,10 % 171,8 % 57]
[(-55) % 342,(-23) % 342,119 % 342,2 % 57]
[107 % 684,(-5) % 684,11 % 684,(-7) % 114]
[7 % 684,(-109) % 684,103 % 684,7 % 114]
verification: h = a * a1 = mult a a1 =
[1 % 1,0 % 1,0 % 1,0 % 1]
[0 % 1,1 % 1,0 % 1,0 % 1]
[0 % 1,0 % 1,1 % 1,0 % 1]
[0 % 1,0 % 1,0 % 1,1 % 1]
test: h == identity =
True
z = a1 * b = mult a1 b =
[2 % 1]
[(-12) % 1]
[(-4) % 1]
[1 % 1]
test: z == x =
True


## J

Implementation:

cramer=:4 :0  A=. x [ b=. y  det=. -/ .*  A %~&det (i.#A) b"_[]}&.|:"0 2 A)

A=: _&".;._2]t=: 0 :0  2 -1  5  1  3  2  2 -6  1  3  3 -1  5 -2 -3  3) b=: _3 _32 _47 49

   A cramer b2 _12 _4 1

## Java

Supports double data type. A more robust solution would support arbitrary precision integers, arbitrary precision decimals, arbitrary precision rationals, or even arbitrary precision algebraic numbers.

 import java.util.ArrayList;import java.util.Arrays;import java.util.List; public class CramersRule {     public static void main(String[] args) {        Matrix mat = new Matrix(Arrays.asList(2d, -1d, 5d, 1d),                                 Arrays.asList(3d, 2d, 2d, -6d),                                 Arrays.asList(1d, 3d, 3d, -1d),                                Arrays.asList(5d, -2d, -3d, 3d));        List<Double> b = Arrays.asList(-3d, -32d, -47d, 49d);        System.out.println("Solution = " + cramersRule(mat, b));    }     private static List<Double> cramersRule(Matrix matrix, List<Double> b) {        double denominator = matrix.determinant();        List<Double> result = new ArrayList<>();        for ( int i = 0 ; i < b.size() ; i++ ) {            result.add(matrix.replaceColumn(b, i).determinant() / denominator);        }        return result;    }     private static class Matrix {         private List<List<Double>> matrix;         @Override        public String toString() {            return matrix.toString();        }         @SafeVarargs        public Matrix(List<Double> ... lists) {            matrix = new ArrayList<>();            for ( List<Double> list : lists) {                matrix.add(list);            }        }         public Matrix(List<List<Double>> mat) {            matrix = mat;        }         public double determinant() {            if ( matrix.size() == 1 ) {                return get(0, 0);            }            if ( matrix.size() == 2 ) {                return get(0, 0) * get(1, 1) - get(0, 1) * get(1, 0);            }            double sum = 0;            double sign = 1;            for ( int i = 0 ; i < matrix.size() ; i++ ) {                sum += sign * get(0, i) * coFactor(0, i).determinant();                sign *= -1;            }            return sum;        }         private Matrix coFactor(int row, int col) {            List<List<Double>> mat = new ArrayList<>();            for ( int i = 0 ; i < matrix.size() ; i++ ) {                if ( i == row ) {                    continue;                }                List<Double> list = new ArrayList<>();                for ( int j = 0 ; j < matrix.size() ; j++ ) {                    if ( j == col ) {                        continue;                    }                    list.add(get(i, j));                }                mat.add(list);            }            return new Matrix(mat);        }         private Matrix replaceColumn(List<Double> b, int column) {            List<List<Double>> mat = new ArrayList<>();            for ( int row = 0 ; row < matrix.size() ; row++ ) {                List<Double> list = new ArrayList<>();                for ( int col = 0 ; col < matrix.size() ; col++ ) {                    double value = get(row, col);                    if ( col == column ) {                        value = b.get(row);                    }                    list.add(value);                }                mat.add(list);            }            return new Matrix(mat);        }         private double get(int row, int col) {            return matrix.get(row).get(col);        }     } }
Output:
Solution = [2.0, -12.0, -4.0, 1.0]


## JavaScript

 var matrix = [	[2, -1,  5,  1],	[3,  2,  2, -6],	[1,  3,  3, -1],	[5, -2, -3,  3]];var freeTerms = [-3, -32, -47, 49]; var result = cramersRule(matrix,freeTerms);console.log(result); /** * 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) {	var det = detr(matrix),		returnArray = [],		i,		tmpMatrix; 	for(i=0; i < matrix.length; i++) {		var 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) {	var tmpMatrix = clone(matrix),		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) {	var ret = 1,		k,		A=clone(m),		n=m.length,		alpha; 	for(var j =0; j < n-1; j++) {		k=j;		for(i=j+1;i<n;i++) { if(Math.abs(A[i][j]) > Math.abs(A[k][j])) { k = i; } }		if(k !== j) {		    temp = A[k]; A[k] = A[j]; A[j] = temp;		    ret *= -1;		}		Aj = A[j];		for(i=j+1;i<n;i++) {			Ai = A[i];            alpha = Ai[j]/Aj[j];            for(k=j+1;k<n-1;k+=2) {                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(function(a){return a.slice();});}
Output:
[ 2, -12, -4, 1 ]

## Julia

Works with: Julia version 0.6
function cramersolve(A::Matrix, b::Vector)    return collect(begin B = copy(A); B[:, i] = b; det(B) end for i in eachindex(b)) ./ det(A)end A = [2 -1  5  1     3  2  2 -6     1  3  3 -1     5 -2 -3  3] b = [-3, -32, -47, 49] @show cramersolve(A, b)
Output:
cramersolve(A, b) = [2.0, -12.0, -4.0, 1.0]

Note that it is entirely impractical to use Cramer's rule in this situation. It would be much better to use the built-in operator for solving linear systems. Assuming that the coefficient matrix and constant vector are defined as before, the solution vector is given by:

@show A \ b

## Kotlin

As in the case of the Matrix arithmetic task, I've used the Johnson-Trotter permutations generator to assist with the calculation of the determinants for the various matrices:

// version 1.1.3 typealias Vector = DoubleArraytypealias Matrix = Array<Vector> fun johnsonTrotter(n: Int): Pair<List<IntArray>, List<Int>> {    val p = IntArray(n) { it }  // permutation    val q = IntArray(n) { it }  // inverse permutation    val d = IntArray(n) { -1 }  // direction = 1 or -1    var sign = 1    val perms = mutableListOf<IntArray>()    val signs = mutableListOf<Int>()     fun permute(k: Int) {        if (k >= n) {            perms.add(p.copyOf())            signs.add(sign)            sign *= -1            return        }        permute(k + 1)        for (i in 0 until k) {            val z = p[q[k] + d[k]]            p[q[k]] = z            p[q[k] + d[k]] = k            q[z] = q[k]            q[k] += d[k]            permute(k + 1)        }        d[k] *= -1    }     permute(0)    return perms to signs} fun determinant(m: Matrix): Double {    val (sigmas, signs) = johnsonTrotter(m.size)    var sum = 0.0    for ((i, sigma) in sigmas.withIndex()) {        var prod = 1.0        for ((j, s) in sigma.withIndex()) prod *= m[j][s]        sum += signs[i] * prod    }    return sum} fun cramer(m: Matrix, d: Vector): Vector {    val divisor = determinant(m)    val numerators = Array(m.size) { Matrix(m.size) { m[it].copyOf() } }    val v = Vector(m.size)    for (i in 0 until m.size) {        for (j in 0 until m.size) numerators[i][j][i] = d[j]    }    for (i in 0 until m.size) v[i] = determinant(numerators[i]) / divisor    return v} fun main(args: Array<String>) {    val m = arrayOf(        doubleArrayOf(2.0, -1.0,  5.0,  1.0),        doubleArrayOf(3.0,  2.0,  2.0, -6.0),        doubleArrayOf(1.0,  3.0,  3.0, -1.0),        doubleArrayOf(5.0, -2.0, -3.0,  3.0)    )    val d = doubleArrayOf(-3.0, -32.0, -47.0, 49.0)    val (w, x, y, z) = cramer(m, d)    println("w = $w, x =$x, y = $y, z =$z")}
Output:
w = 2.0, x = -12.0, y = -4.0, z = 1.0


## Lua

local matrix = require "matrix" -- https://github.com/davidm/lua-matrix local function cramer(mat, vec)  -- Check if matrix is quadratic  assert(#mat == #mat, "Matrix is not square!")  -- Check if vector has the same size of the matrix  assert(#mat == #vec, "Vector has not the same size of the matrix!")   local size = #mat  local main_det = matrix.det(mat)   local aux_mats = {}  local dets = {}  local result = {}  for i = 1, size do    -- Construct the auxiliary matrixes    aux_mats[i] = matrix.copy(mat)    for j = 1, size do      aux_mats[i][j][i] = vec[j]    end     -- Calculate the auxiliary determinants    dets[i] = matrix.det(aux_mats[i])     -- Calculate results    result[i] = dets[i]/main_det  end   return resultend ----------------------------------------------- local A = {{ 2, -1,  5,  1},           { 3,  2,  2, -6},           { 1,  3,  3, -1},           { 5, -2, -3,  3}}local b = {-3, -32, -47, 49} local result = cramer(A, b)print("Result: " .. table.concat(result, ", "))
Output:
Result: 2, -12, -4, 1

## Maple

with(LinearAlgebra):cramer:=proc(A,B)  local n,d,X,V,i;  n:=upperbound(A,2);  d:=Determinant(A);  X:=Vector(n,0);  for i from 1 to n do    V:=A(1..-1,i);    A(1..-1,i):=B;    X[i]:=Determinant(A)/d;    A(1..-1,i):=V;  od;  X;end: A:=Matrix([[2,-1,5,1],[3,2,2,-6],[1,3,3,-1],[5,-2,-3,3]]):B:=Vector([-3,-32,-47,49]):printf("%a",cramer(A,B));
Output:
Vector(4, [2,-12,-4,1])

## Mathematica

crule[m_, b_] := Module[{d = Det[m], a},  Table[a = m; a[[All, k]] = b; Det[a]/d, {k, Length[m]}]] crule[{  {2, -1, 5, 1},  {3, 2, 2, -6},  {1, 3, 3, -1},  {5, -2, -3, 3} } , {-3, -32, -47, 49}]
Output:
{2,-12,-4,1}

## Maxima

 (%i1) eqns: [ 2*w-x+5*y+z=-3, 3*w+2*x+2*y-6*z=-32, w+3*x+3*y-z=-47, 5*w-2*x-3*y+3*z=49];(%o1) [z + 5 y - x + 2 w = - 3, (- 6 z) + 2 y + 2 x + 3 w = - 32,                       (- z) + 3 y + 3 x + w = - 47, 3 z - 3 y - 2 x + 5 w = 49](%i2) A: augcoefmatrix (eqns, [w,x,y,z]);                          [ 2  - 1   5    1    3   ]                          [                        ]                          [ 3   2    2   - 6   32  ](%o2)                     [                        ]                          [ 1   3    3   - 1   47  ]                          [                        ]                          [ 5  - 2  - 3   3   - 49 ](%i3) C: coefmatrix(eqns, [w,x,y,z]);                             [ 2  - 1   5    1  ]                             [                  ]                             [ 3   2    2   - 6 ](%o3)                        [                  ]                             [ 1   3    3   - 1 ]                             [                  ]                             [ 5  - 2  - 3   3  ](%i4) c[n]:= (-1)^(n+1) * determinant (submatrix (A,n))/determinant (C);                            n + 1                       (- 1)      determinant(submatrix(A, n))(%o4)            c  := ---------------------------------------                  n                determinant(C)(%i5) makelist (c[n],n,1,4);(%o5)                          [2, - 12, - 4, 1](%i6) linsolve(eqns, [w,x,y,z]);(%o6)                  [w = 2, x = - 12, y = - 4, z = 1]

## PARI/GP

 M = [2,-1,5,1;3,2,2,-6;1,3,3,-1;5,-2,-3,3];V = Col([-3,-32,-47,49]); matadjoint(M) * V / matdet(M)

Output:

[2, -12, -4, 1]~

use Math::Matrix; sub cramers_rule {    my ($A,$terms) = @_;    my @solutions;    my $det =$A->determinant;    foreach my $i (0 ..$#{$A}) { my$Ai = $A->clone; foreach my$j (0 .. $#{$terms}) {            $Ai->[$j][$i] =$terms->[$j]; } push @solutions,$Ai->determinant / $det; } @solutions;} my$matrix = Math::Matrix->new(    [2, -1,  5,  1],    [3,  2,  2, -6],    [1,  3,  3, -1],    [5, -2, -3,  3],); my $free_terms = [-3, -32, -47, 49];my ($w, $x,$y, $z) = cramers_rule($matrix, $free_terms); print "w =$w\n";print "x = $x\n";print "y =$y\n";print "z = $z\n"; Output: w = 2 x = -12 y = -4 z = 1  ## Phix Translation of: C The copy-on-write semantics of Phix really shine here; because there is no explicit return/re-assign, you can treat parameters as a private workspace, confident in the knowledge that the updated version will be quietly discarded; all the copying and freeing of the C version is automatic/unnecessary here. constant inf = 1e300*1e300, nan = -(inf/inf) function det(sequence a)atom res = 1 for j=1 to length(a) do integer i_max = j for i=j+1 to length(a) do if a[i][j] > a[i_max][j] then i_max = i end if end for if i_max != j then {a[i_max],a[j]} = {a[j],a[i_max]} res *= -1 end if if abs(a[j][j]) < 1e-12 then puts(1,"Singular matrix!") return nan end if for i=j+1 to length(a) do atom mult = -a[i][j] / a[j][j] for k=1 to length(a) do a[i][k] += mult * a[j][k] end for end for end for for i=1 to length(a) do res *= a[i][i] end for return resend function function cramer_solve(sequence a, atom det_a, sequence b, integer var) for i=1 to length(a) do a[i][var] = b[i] end for return det(a)/det_aend function sequence a = {{2,-1, 5, 1}, {3, 2, 2,-6}, {1, 3, 3,-1}, {5,-2,-3, 3}}, b = {-3,-32,-47,49}integer det_a = det(a)for i=1 to length(a) do printf(1, "%7.3f\n", cramer_solve(a, det_a, b, i))end for Output:  2.000 -12.000 -4.000 1.000  ## Prolog Works with: GNU Prolog removeElement([_|Tail], 0, Tail).removeElement([Head|Tail], J, [Head|X]) :- J_2 is J - 1, removeElement(Tail, J_2, X). removeColumn([], _, []).removeColumn([Matrix_head|Matrix_tail], J, [X|Y]) :- removeElement(Matrix_head, J, X), removeColumn(Matrix_tail, J, Y). removeRow([_|Matrix_tail], 0, Matrix_tail).removeRow([Matrix_head|Matrix_tail], I, [Matrix_head|X]) :- I_2 is I - 1, removeRow(Matrix_tail, I_2, X). cofactor(Matrix, I, J, X) :- removeRow(Matrix, I, Matrix_2), removeColumn(Matrix_2, J, Matrix_3), det(Matrix_3, Y), X is (-1) ** (I + J) * Y. det_summand(_, _, [], 0).det_summand(Matrix, J, B, X) :- B = [B_head|B_tail], cofactor(Matrix, 0, J, Z), J_2 is J + 1, det_summand(Matrix, J_2, B_tail, Y), X is B_head * Z + Y. det([[X]], X).det(Matrix, X) :- Matrix = [Matrix_head|_], det_summand(Matrix, 0, Matrix_head, X). replaceElement([_|Tail], 0, New, [New|Tail]).replaceElement([Head|Tail], J, New, [Head|Y]) :- J_2 is J - 1, replaceElement(Tail, J_2, New, Y). replaceColumn([], _, _, []).replaceColumn([Matrix_head|Matrix_tail], J, [Column_head|Column_tail], [X|Y]) :- replaceElement(Matrix_head, J, Column_head, X), replaceColumn(Matrix_tail, J, Column_tail, Y). cramerElements(_, B, L, []) :- length(B, L).cramerElements(A, B, J, [X_J|Others]) :- replaceColumn(A, J, B, A_J), det(A_J, Det_A_J), det(A, Det_A), X_J is Det_A_J / Det_A, J_2 is J + 1, cramerElements(A, B, J_2, Others). cramer(A, B, X) :- cramerElements(A, B, 0, X). results(X) :- A = [ [2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3] ], B = [-3, -32, -47, 49], cramer(A, B, X). Output: | ?- results(X). X = [2.0,-12.0,-4.0,1.0] ? yes  ## Python  # a simple implementation using numpyfrom numpy import linalg A=[[2,-1,5,1],[3,2,2,-6],[1,3,3,-1],[5,-2,-3,3]]B=[-3,-32,-47,49]C=[[2,-1,5,1],[3,2,2,-6],[1,3,3,-1],[5,-2,-3,3]]X=[]for i in range(0,len(B)): for j in range(0,len(B)): C[j][i]=B[j] if i>0: C[j][i-1]=A[j][i-1] X.append(round(linalg.det(C)/linalg.det(A),1)) print('w=%s'%X,'x=%s'%X,'y=%s'%X,'z=%s'%X)  Output: w=2.0 x=-12.0 y=-4.0 z=1.0  ## Racket  This example is incorrect. Please fix the code and remove this message.Details: Doesn't really use Cramer's rule #lang typed/racket(require math/matrix) (define A (matrix [[2 -1 5 1] [3 2 2 -6] [1 3 3 -1] [5 -2 -3 3]])) (define B (col-matrix [ -3 -32 -47 49])) (matrix->vector (matrix-solve A B)) Output: '#(2 -12 -4 1) ## Raku (formerly Perl 6) sub det(@matrix) { my @a = @matrix.map: { [|$_] };    my $sign = +1; my$pivot = 1;    for ^@a -> $k { my @r = ($k+1 .. @a.end);      my $previous-pivot =$pivot;      if 0 == ($pivot = @a[$k][$k]) { (my$s = @r.first: { @a[$_][$k] != 0 }) // return 0;        (@a[$s],@a[$k]) = (@a[$k], @a[$s]);        my $pivot = @a[$k][$k];$sign = -$sign; } for @r X @r -> ($i, $j) { ((@a[$i][$j] *=$pivot) -= @a[$i][$k]*@a[$k][$j]) /= $previous-pivot; } }$sign * $pivot} sub cramers_rule(@A, @terms) { gather for ^@A ->$i {        my @Ai = @A.map: { [|$_] }; for ^@terms ->$j {            @Ai[$j][$i] = @terms[$j]; } take det(@Ai); } »/» det(@A);} my @matrix = ( [2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3],); my @free_terms = (-3, -32, -47, 49);my ($w, $x,$y, $z) = |cramers_rule(@matrix, @free_terms); say "w =$w";say "x = $x";say "y =$y";say "z = $z"; Output: w = 2 x = -12 y = -4 z = 1  ## REXX /* REXX Use Cramer's rule to compute solutions of given linear equations */Numeric Digits 20names='w x y z'M=' 2 -1 5 1', ' 3 2 2 -6', ' 1 3 3 -1', ' 5 -2 -3 3'v=' -3', '-32', '-47', ' 49'Call mk_mat(m) /* M -> a.i.j */Do j=1 To dim /* Show the input */ ol='' Do i=1 To dim ol=ol format(a.i.j,6) End ol=ol format(word(v,j),6) Say ol EndSay copies('-',35) d=det(m) /* denominator determinant */ Do k=1 To dim /* construct nominator matrix */ Do j=1 To dim Do i=1 To dim If i=k Then b.i.j=word(v,j) Else b.i.j=a.i.j End End Call show_b d.k=det(mk_str()) /* numerator determinant */ Say word(names,k) '=' d.k/d /* compute value of variable k */ EndExit mk_mat: Procedure Expose a. dim /* Turn list into matrix a.i.j */ Parse Arg list dim=sqrt(words(list)) k=0 Do j=1 To dim Do i=1 To dim k=k+1 a.i.j=word(list,k) End End Return mk_str: Procedure Expose b. dim /* Turn matrix b.i.j into list */ str=''Do j=1 To dim Do i=1 To dim str=str b.i.j End EndReturn str show_b: Procedure Expose b. dim /* show numerator matrix */ do j=1 To dim ol='' Do i=1 To dim ol=ol format(b.i.j,6) end Call dbg ol end Return det: Procedure /* compute determinant */Parse Arg listn=words(list)call dbg 'det:' listdo dim=1 To 10 If dim**2=n Then Leave Endcall dbg 'dim='dimIf dim=2 Then Do det=word(list,1)*word(list,4)-word(list,2)*word(list,3) call dbg 'det=>'det Return det Endk=0Do j=1 To dim Do i=1 To dim k=k+1 a.i.j=word(list,k) End EndDo j=1 To dim ol=j Do i=1 To dim ol=ol format(a.i.j,6) End call dbg ol Enddet=0Do i=1 To dim ol='' Do j=2 To dim Do ii=1 To dim If ii<>i Then ol=ol a.ii.j End End call dbg 'i='i 'ol='ol If i//2 Then det=det+a.i.1*det(ol) Else det=det-a.i.1*det(ol) EndCall dbg 'det=>>>'detReturn detsqrt: Procedure/* REXX **************************************************************** EXEC to calculate the square root of a = 2 with high precision**********************************************************************/ Parse Arg x,prec If prec<9 Then prec=9 prec1=2*prec eps=10**(-prec1) k = 1 Numeric Digits 3 r0= x r = 1 Do i=1 By 1 Until r=r0 | (abs(r*r-x)<eps) r0 = r r = (r + x/r) / 2 k = min(prec1,2*k) Numeric Digits (k + 5) End Numeric Digits prec r=r+0 Return r dbg: Return Output:  2 -1 5 1 -3 3 2 2 -6 -32 1 3 3 -1 -47 5 -2 -3 3 49 ----------------------------------- w = 2 x = -12 y = -4 z = 1 ## Ruby require 'matrix' def cramers_rule(a, terms) raise ArgumentError, " Matrix not square" unless a.square? cols = a.to_a.transpose cols.each_index.map do |i| c = cols.dup c[i] = terms Matrix.columns(c).det / a.det endend matrix = Matrix[ [2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3],] vector = [-3, -32, -47, 49]puts cramers_rule(matrix, vector) Output: 2 -12 -4 1  ## Rust use std::ops::{Index, IndexMut}; fn main() { let m = matrix( vec![ 2., -1., 5., 1., 3., 2., 2., -6., 1., 3., 3., -1., 5., -2., -3., 3., ], 4, ); let mm = m.solve(&vec![-3., -32., -47., 49.]); println!("{:?}", mm);} #[derive(Clone)]struct Matrix { elts: Vec<f64>, dim: usize,} impl Matrix { // Compute determinant using cofactor method // Using Gaussian elimination would have been more efficient, but it also solves the linear // system, so… fn det(&self) -> f64 { match self.dim { 0 => 0., 1 => self, 2 => self * self - self * self, d => { let mut acc = 0.; let mut signature = 1.; for k in 0..d { acc += signature * self[k] * self.comatrix(0, k).det(); signature *= -1. } acc } } } // Solve linear systems using Cramer's method fn solve(&self, target: &Vec<f64>) -> Vec<f64> { let mut solution: Vec<f64> = vec![0.; self.dim]; let denominator = self.det(); for j in 0..self.dim { let mut mm = self.clone(); for i in 0..self.dim { mm[i][j] = target[i] } solution[j] = mm.det() / denominator } solution } // Compute the cofactor matrix for determinant computations fn comatrix(&self, k: usize, l: usize) -> Matrix { let mut v: Vec<f64> = vec![]; for i in 0..self.dim { for j in 0..self.dim { if i != k && j != l { v.push(self[i][j]) } } } matrix(v, self.dim - 1) }} fn matrix(elts: Vec<f64>, dim: usize) -> Matrix { assert_eq!(elts.len(), dim * dim); Matrix { elts, dim }} impl Index<usize> for Matrix { type Output = [f64]; fn index(&self, i: usize) -> &Self::Output { let m = self.dim; &self.elts[m * i..m * (i + 1)] }} impl IndexMut<usize> for Matrix { fn index_mut(&mut self, i: usize) -> &mut Self::Output { let m = self.dim; &mut self.elts[m * i..m * (i + 1)] }} Which outputs: [2.0, -12.0, -4.0, 1.0] ## Sidef func cramers_rule(A, terms) { gather { for i in ^A { var Ai = A.map{.map{_}} for j in ^terms { Ai[j][i] = terms[j] } take(Ai.det) } } »/» A.det} var matrix = [ [2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3],] var free_terms = [-3, -32, -47, 49]var (w, x, y, z) = cramers_rule(matrix, free_terms)... say "w = #{w}"say "x = #{x}"say "y = #{y}"say "z = #{z}" Output: w = 2 x = -12 y = -4 z = 1  ## Tcl  package require math::linearalgebranamespace path ::math::linearalgebra # Setting matrix to variable A and size to nset A [list { 2 -1 5 1} { 3 2 2 -6} { 1 3 3 -1} { 5 -2 -3 3}]set n [llength$A]# Setting right side of equationset right {-3 -32 -47 49} # Calculating determinant of Aset detA [det $A] # Apply Cramer's rulefor {set i 0} {$i < $n} {incr i} { set tmp$A                    ;# copy A to tmp  setcol tmp $i$right          ;# replace column i with right side vector  set detTmp [det $tmp] ;# calculate determinant of tmp set v [expr$detTmp / $detA] ;# divide two determinants puts [format "%0.4f"$v]      ;# format and display result}
Output:
2.0000
-12.0000
-4.0000
1.0000


## Visual Basic .NET

Translation of: C#
Imports System.Runtime.CompilerServicesImports System.Linq.Enumerable Module Module1    <Extension()>    Function DelimitWith(Of T)(source As IEnumerable(Of T), Optional seperator As String = " ") As String        Return String.Join(seperator, source)    End Function     Private Class SubMatrix        Private ReadOnly source As Integer(,)        Private ReadOnly prev As SubMatrix        Private ReadOnly replaceColumn As Integer()         Public Sub New(source As Integer(,), replaceColumn As Integer())            Me.source = source            Me.replaceColumn = replaceColumn            prev = Nothing            ColumnIndex = -1            Size = replaceColumn.Length        End Sub         Public Sub New(prev As SubMatrix, Optional deletedColumnIndex As Integer = -1)            source = Nothing            replaceColumn = Nothing            Me.prev = prev            ColumnIndex = deletedColumnIndex            Size = prev.Size - 1        End Sub         Public Property ColumnIndex As Integer        Public ReadOnly Property Size As Integer         Default Public ReadOnly Property Index(row As Integer, column As Integer) As Integer            Get                If Not IsNothing(source) Then                    Return If(column = ColumnIndex, replaceColumn(row), source(row, column))                Else                    Return prev(row + 1, If(column < ColumnIndex, column, column + 1))                End If            End Get        End Property         Public Function Det() As Integer            If Size = 1 Then Return Me(0, 0)            If Size = 2 Then Return Me(0, 0) * Me(1, 1) - Me(0, 1) * Me(1, 0)            Dim m As New SubMatrix(Me)            Dim detVal = 0            Dim sign = 1            For c = 0 To Size - 1                m.ColumnIndex = c                Dim d = m.Det()                detVal += Me(0, c) * d * sign                sign = -sign            Next            Return detVal        End Function         Public Sub Print()            For r = 0 To Size - 1                Dim rl = r                Console.WriteLine(Range(0, Size).Select(Function(c) Me(rl, c)).DelimitWith(", "))            Next            Console.WriteLine()        End Sub    End Class     Private Function Solve(matrix As SubMatrix) As Integer()        Dim det = matrix.Det()        If det = 0 Then Throw New ArgumentException("The determinant is zero.")         Dim answer(matrix.Size - 1) As Integer        For i = 0 To matrix.Size - 1            matrix.ColumnIndex = i            answer(i) = matrix.Det() / det        Next        Return answer    End Function     Public Function SolveCramer(equations As Integer()()) As Integer()        Dim size = equations.Length        If equations.Any(Function(eq) eq.Length <> size + 1) Then Throw New ArgumentException(\$"Each equation must have {size + 1} terms.")        Dim matrix(size - 1, size - 1) As Integer        Dim column(size - 1) As Integer        For r = 0 To size - 1            column(r) = equations(r)(size)            For c = 0 To size - 1                matrix(r, c) = equations(r)(c)            Next        Next        Return Solve(New SubMatrix(matrix, column))    End Function     Sub Main()        Dim equations = {            ({2, -1, 5, 1, -3}),            ({3, 2, 2, -6, -32}),            ({1, 3, 3, -1, -47}),            ({5, -2, -3, 3, 49})        }        Dim solution = SolveCramer(equations)        Console.WriteLine(solution.DelimitWith(", "))    End Sub End Module
Output:
2, -12, -4, 1

## zkl

Using the GNU Scientific Library, we define the values:

var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)A:=GSL.Matrix(4,4).set(2,-1, 5, 1,		       3, 2, 2,-6,		       1, 3, 3,-1,		       5,-2,-3, 3);b:=GSL.Vector(4).set(-3,-32,-47,49);

First, just let GSL solve:

A.AxEQb(b).format().println();

Actually implement Cramer's rule:

Translation of: Julia
fcn cramersRule(A,b){   b.len().pump(GSL.Vector(b.len()),'wrap(i){ // put calculations into new Vector      A.copy().setColumn(i,b).det();   }).close()/A.det();}cramersRule(A,b).format().println();
Output:
2.00,-12.00,-4.00,1.00