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Gauss-Jordan matrix inversion

;Task: Invert matrix '''A''' using Gauss-Jordan method.

'''A''' being an '''n''' by '''n''' matrix.

## C#

```
using System;

namespace Rosetta
{
internal class Vector
{
private double[] b;

internal Vector(int rows)
{
this.rows = rows;
b = new double[rows];
}

internal Vector(double[] initArray)
{
b = (double[])initArray.Clone();
rows = b.Length;
}

internal Vector Clone()
{
Vector v = new Vector(b);
return v;
}

internal double this[int row]
{
get { return b[row]; }
set { b[row] = value; }
}

internal void SwapRows(int r1, int r2)
{
if (r1 == r2) return;
double tmp = b[r1];
b[r1] = b[r2];
b[r2] = tmp;
}

internal double norm(double[] weights)
{
double sum = 0;
for (int i = 0; i < rows; i++)
{
double d = b[i] * weights[i];
sum +=  d*d;
}
return Math.Sqrt(sum);
}

internal void print()
{
for (int i = 0; i < rows; i++)
Console.WriteLine(b[i]);
Console.WriteLine();
}

public static Vector operator-(Vector lhs, Vector rhs)
{
Vector v = new Vector(lhs.rows);
for (int i = 0; i < lhs.rows; i++)
v[i] = lhs[i] - rhs[i];
return v;
}
}

class Matrix
{
private double[] b;

internal Matrix(int rows, int cols)
{
this.rows = rows;
this.cols = cols;
b = new double[rows * cols];
}

internal Matrix(int size)
{
this.rows = size;
this.cols = size;
b = new double[rows * cols];
for (int i = 0; i < size; i++)
this[i, i] = 1;
}

internal Matrix(int rows, int cols, double[] initArray)
{
this.rows = rows;
this.cols = cols;
b = (double[])initArray.Clone();
if (b.Length != rows * cols) throw new Exception("bad init array");
}

internal double this[int row, int col]
{
get { return b[row * cols + col]; }
set { b[row * cols + col] = value; }
}

public static Vector operator*(Matrix lhs, Vector rhs)
{
if (lhs.cols != rhs.rows) throw new Exception("I can't multiply matrix by vector");
Vector v = new Vector(lhs.rows);
for (int i = 0; i < lhs.rows; i++)
{
double sum = 0;
for (int j = 0; j < rhs.rows; j++)
sum += lhs[i,j]*rhs[j];
v[i] = sum;
}
return v;
}

internal void SwapRows(int r1, int r2)
{
if (r1 == r2) return;
int firstR1 = r1 * cols;
int firstR2 = r2 * cols;
for (int i = 0; i < cols; i++)
{
double tmp = b[firstR1 + i];
b[firstR1 + i] = b[firstR2 + i];
b[firstR2 + i] = tmp;
}
}

//with partial pivot
internal bool InvPartial()
{
const double Eps = 1e-12;
if (rows != cols) throw new Exception("rows != cols for Inv");
Matrix M = new Matrix(rows); //unitary
for (int diag = 0; diag < rows; diag++)
{
int max_row = diag;
double max_val = Math.Abs(this[diag, diag]);
double d;
for (int row = diag + 1; row < rows; row++)
if ((d = Math.Abs(this[row, diag])) > max_val)
{
max_row = row;
max_val = d;
}
if (max_val <= Eps) return false;
SwapRows(diag, max_row);
M.SwapRows(diag, max_row);
double invd = 1 / this[diag, diag];
for (int col = diag; col < cols; col++)
{
this[diag, col] *= invd;
}
for (int col = 0; col < cols; col++)
{
M[diag, col] *= invd;
}
for (int row = 0; row < rows; row++)
{
d = this[row, diag];
if (row != diag)
{
for (int col = diag; col < this.cols; col++)
{
this[row, col] -= d * this[diag, col];
}
for (int col = 0; col < this.cols; col++)
{
M[row, col] -= d * M[diag, col];
}
}
}
}
b = M.b;
return true;
}

internal void print()
{
for (int i = 0; i < rows; i++)
{
for (int j = 0; j < cols; j++)
Console.Write(this[i,j].ToString()+"  ");
Console.WriteLine();
}
}
}
}

```
```
using System;

namespace Rosetta
{
class Program
{
static void Main(string[] args)
{
Matrix M = new Matrix(4, 4, new double[] { -1, -2, 3, 2, -4, -1, 6, 2, 7, -8, 9, 1, 1, -2, 1, 3 });
M.InvPartial();
M.print();
}
}
}

```

{{out}}

```
-0.913043478260869  0.246376811594203  0.0942028985507246  0.413043478260869
-1.65217391304348  0.652173913043478  0.0434782608695652  0.652173913043478
-0.695652173913043  0.36231884057971  0.0797101449275362  0.195652173913043
-0.565217391304348  0.231884057971014  -0.0289855072463768  0.565217391304348

```

## Go

{{trans|Kotlin}}

```package main

import "fmt"

type vector = []float64
type matrix []vector

func (m matrix) inverse() matrix {
le := len(m)
for _, v := range m {
if len(v) != le {
panic("Not a square matrix")
}
}
aug := make(matrix, le)
for i := 0; i < le; i++ {
aug[i] = make(vector, 2*le)
copy(aug[i], m[i])
// augment by identity matrix to right
aug[i][i+le] = 1
}
aug.toReducedRowEchelonForm()
inv := make(matrix, le)
// remove identity matrix to left
for i := 0; i < le; i++ {
inv[i] = make(vector, le)
copy(inv[i], aug[i][le:])
}
return inv
}

// note: this mutates the matrix in place
func (m matrix) toReducedRowEchelonForm() {
rowCount, colCount := len(m), len(m[0])
for r := 0; r < rowCount; r++ {
return
}
i := r

i++
if rowCount == i {
i = r
return
}
}
}

m[i], m[r] = m[r], m[i]
if div := m[r][lead]; div != 0 {
for j := 0; j < colCount; j++ {
m[r][j] /= div
}
}

for k := 0; k < rowCount; k++ {
if k != r {
for j := 0; j < colCount; j++ {
m[k][j] -= m[r][j] * mult
}
}
}
}
}

func (m matrix) print(title string) {
fmt.Println(title)
for _, v := range m {
fmt.Printf("% f\n", v)
}
fmt.Println()
}

func main() {
a := matrix{{1, 2, 3}, {4, 1, 6}, {7, 8, 9}}
a.inverse().print("Inverse of A is:\n")

b := matrix{{2, -1, 0}, {-1, 2, -1}, {0, -1, 2}}
b.inverse().print("Inverse of B is:\n")
}
```

{{out}}

```
Inverse of A is:

[-0.812500  0.125000  0.187500]
[ 0.125000 -0.250000  0.125000]
[ 0.520833  0.125000 -0.145833]

Inverse of B is:

[ 0.750000  0.500000  0.250000]
[ 0.500000  1.000000  0.500000]
[ 0.250000  0.500000  0.750000]

```

## J

'''Solution:'''

Uses Gauss-Jordan implementation (as described in [[Reduced_row_echelon_form#J]]) to find reduced row echelon form of the matrix after augmenting with an identity matrix.

```require 'math/misc/linear'
augmentR_I1=: ,. e.@i.@#       NB. augment matrix on the right with its Identity matrix
matrix_invGJ=: # }."1 [: gauss_jordan@augmentR_I1
```

'''Usage:'''

```   ]A =: 1 2 3, 4 1 6,: 7 8 9
1 2 3
4 1 6
7 8 9
matrix_invGJ A
_0.8125 0.125    0.1875
0.125 _0.25     0.125
0.520833 0.125 _0.145833
```

## Julia

{{works with|Julia|0.6}}

'''Built-in''' LAPACK-based linear solver uses partial-pivoted Gauss elimination):

```A = [1 2 3; 4 1 6; 7 8 9]
@show I / A
@show inv(A)
```

'''Native implementation''':

```function gaussjordan(A::Matrix)
size(A, 1) == size(A, 2) || throw(ArgumentError("A must be squared"))
n = size(A, 1)
M = [convert(Matrix{float(eltype(A))}, A) I]
i = 1
local tmp = Vector{eltype(M)}(2n)
# forward
while i ≤ n
if M[i, i] ≈ 0.0
local j = i + 1
while j ≤ n && M[j, i] ≈ 0.0
j += 1
end
if j ≤ n
tmp     .= M[i, :]
M[i, :] .= M[j, :]
M[j, :] .= tmp
else
throw(ArgumentError("matrix is singular, cannot compute the inverse"))
end
end
for j in (i + 1):n
M[j, :] .-= M[j, i] / M[i, i] .* M[i, :]
end
i += 1
end
i = n
# backward
while i ≥ 1
if M[i, i] ≈ 0.0
local j = i - 1
while j ≥ 1 && M[j, i] ≈ 0.0
j -= 1
end
if j ≥ 1
tmp     .= M[i, :]
M[i, :] .= M[j, :]
M[j, :] .= tmp
else
throw(ArgumentError("matrix is singular, cannot compute the inverse"))
end
end
for j in (i - 1):-1:1
M[j, :] .-= M[j, i] / M[i, i] .* M[i, :]
end
i -= 1
end
M ./= diag(M) # normalize
return M[:, n+1:2n]
end

@show gaussjordan(A)
@assert gaussjordan(A) ≈ inv(A)

A = rand(10, 10)
@assert gaussjordan(A) ≈ inv(A)
```

{{out}}

```I / A = [-0.8125 0.125 0.1875; 0.125 -0.25 0.125; 0.520833 0.125 -0.145833]
inv(A) = [-0.8125 0.125 0.1875; 0.125 -0.25 0.125; 0.520833 0.125 -0.145833]
gaussjordan(A) = [-0.8125 0.125 0.1875; 0.125 -0.25 0.125; 0.520833 0.125 -0.145833]
```

## Kotlin

This follows the description of Gauss-Jordan elimination in Wikipedia whereby the original square matrix is first augmented to the right by its identity matrix, its reduced row echelon form is then found and finally the identity matrix to the left is removed to leave the inverse of the original square matrix.

```// version 1.2.21

typealias Matrix = Array<DoubleArray>

fun Matrix.inverse(): Matrix {
val len = this.size
require(this.all { it.size == len }) { "Not a square matrix" }
val aug = Array(len) { DoubleArray(2 * len) }
for (i in 0 until len) {
for (j in 0 until len) aug[i][j] = this[i][j]
// augment by identity matrix to right
aug[i][i + len] = 1.0
}
aug.toReducedRowEchelonForm()
val inv = Array(len) { DoubleArray(len) }
// remove identity matrix to left
for (i in 0 until len) {
for (j in len until 2 * len) inv[i][j - len] = aug[i][j]
}
return inv
}

fun Matrix.toReducedRowEchelonForm() {
val rowCount = this.size
val colCount = this[0].size
for (r in 0 until rowCount) {
var i = r

i++
if (rowCount == i) {
i = r
}
}

val temp = this[i]
this[i] = this[r]
this[r] = temp

for (j in 0 until colCount) this[r][j] /= div
}

for (k in 0 until rowCount) {
if (k != r) {
for (j in 0 until colCount) this[k][j] -= this[r][j] * mult
}
}

}
}

fun Matrix.printf(title: String) {
println(title)
val rowCount = this.size
val colCount = this[0].size

for (r in 0 until rowCount) {
for (c in 0 until colCount) {
if (this[r][c] == -0.0) this[r][c] = 0.0  // get rid of negative zeros
print("\${"% 10.6f".format(this[r][c])}  ")
}
println()
}

println()
}

fun main(args: Array<String>) {
val a = arrayOf(
doubleArrayOf(1.0, 2.0, 3.0),
doubleArrayOf(4.0, 1.0, 6.0),
doubleArrayOf(7.0, 8.0, 9.0)
)
a.inverse().printf("Inverse of A is :\n")

val b = arrayOf(
doubleArrayOf( 2.0, -1.0,  0.0),
doubleArrayOf(-1.0,  2.0, -1.0),
doubleArrayOf( 0.0, -1.0,  2.0)
)
b.inverse().printf("Inverse of B is :\n")
}
```

{{out}}

```
Inverse of A is :

-0.812500    0.125000    0.187500
0.125000   -0.250000    0.125000
0.520833    0.125000   -0.145833

Inverse of B is :

0.750000    0.500000    0.250000
0.500000    1.000000    0.500000
0.250000    0.500000    0.750000

```

## Perl

Included code from [[Reduced_row_echelon_form#Perl|Reduced row echelon form]] task.

```sub rref {
our @m; local *m = shift;
@m or return;
my (\$lead, \$rows, \$cols) = (0, scalar(@m), scalar(@{\$m[0]}));

foreach my \$r (0 .. \$rows - 1) {
my \$i = \$r;

{++\$i == \$rows or next;
\$i = \$r;

@m[\$i, \$r] = @m[\$r, \$i];
\$_ /= \$lv foreach @{ \$m[\$r] };

my @mr = @{ \$m[\$r] };
foreach my \$i (0 .. \$rows - 1)
{\$i == \$r and next;
(\$lv, my \$n) = (\$m[\$i][\$lead], -1);
\$_ -= \$lv * \$mr[++\$n] foreach @{ \$m[\$i] };}

}

sub display { join("\n" => map join(" " => map(sprintf("%6.2f", \$_), @\$_)), @{+shift})."\n" }

sub gauss_jordan_invert {
my(@m) = @_;
my \$rows = @m;
my @i = identity(scalar @m);
push @{\$m[\$_]}, @{\$i[\$_]} for 0..\$rows-1;
rref(\@m);
map { splice @\$_, 0, \$rows } @m;
@m;
}

sub identity {
my(\$n) = @_;
map { [ (0) x \$_, 1, (0) x (\$n-1 - \$_) ] } 0..\$n-1
}

my @tests = (
[
[ 2, -1,  0 ],
[-1,  2, -1 ],
[ 0, -1,  2 ]
],
[
[ -1, -2, 3, 2 ],
[ -4, -1, 6, 2 ],
[  7, -8, 9, 1 ],
[  1, -2, 1, 3 ]
],
);

for my \$matrix (@tests) {
print "Original Matrix:\n" . display(\@\$matrix) . "\n";
my @gj = gauss_jordan_invert( @\$matrix );
print "Gauss-Jordan Inverted Matrix:\n" . display(\@gj) . "\n";
my @rt = gauss_jordan_invert( @gj );
print "After round-trip:\n" . display(\@rt) . "\n";} . "\n"
}
```

{{out}}

```Original Matrix:
2.00  -1.00   0.00
-1.00   2.00  -1.00
0.00  -1.00   2.00

Gauss-Jordan Inverted Matrix:
0.75   0.50   0.25
0.50   1.00   0.50
0.25   0.50   0.75

After round-trip:
2.00  -1.00   0.00
-1.00   2.00  -1.00
0.00  -1.00   2.00

Original Matrix:
-1.00  -2.00   3.00   2.00
-4.00  -1.00   6.00   2.00
7.00  -8.00   9.00   1.00
1.00  -2.00   1.00   3.00

Gauss-Jordan Inverted Matrix:
-0.91   0.25   0.09   0.41
-1.65   0.65   0.04   0.65
-0.70   0.36   0.08   0.20
-0.57   0.23  -0.03   0.57

After round-trip:
-1.00  -2.00   3.00   2.00
-4.00  -1.00   6.00   2.00
7.00  -8.00   9.00   1.00
1.00  -2.00   1.00   3.00
```

## Perl 6

{{works with|Rakudo|2019.03.1}} Uses bits and pieces from other tasks, [[Reduced_row_echelon_form#Perl_6|Reduced row echelon form]] primarily.

```sub gauss-jordan-invert (@m where *.&is-square) {
^@m .map: { @m[\$_].append: identity(+@m)[\$_] };
@m.&rref[*]»[+@m .. *];
}

sub is-square (@m) { so @m == all @m[*] }

sub identity (\$n) { [ 1, |(0 xx \$n-1) ], *.rotate(-1) ... *.tail }

# reduced row echelon form (Gauss-Jordan elimination)
sub rref (@m) {
return unless @m;
my (\$lead, \$rows, \$cols) = 0, +@m, +@m[0];

for ^\$rows -> \$r {
\$lead < \$cols or return @m;
my \$i = \$r;
++\$i == \$rows or next;
\$i = \$r;
++\$lead == \$cols and return @m;
}
@m[\$i, \$r] = @m[\$r, \$i] if \$r != \$i;
@m[\$r] »/=» \$lv;
for ^\$rows -> \$n {
next if \$n == \$r;
@m[\$n] »-=» @m[\$r] »*» (@m[\$n;\$lead] // 0);
}
}
@m
}

sub rat-or-int (\$num) {
return \$num unless \$num ~~ Rat;
return \$num.narrow if \$num.narrow.WHAT ~~ Int;
\$num.nude.join: '/';
}

sub say_it (\$message, @array) {
my \$max;
@array.map: {\$max max= max \$_».&rat-or-int.comb(/\S+/)».chars};
say "\n\$message";
\$_».&rat-or-int.fmt(" %{\$max}s").put for @array;
}

sub to-matrix (\$str) { [\$str.split(';').map(*.words.Array)] }

my @tests =
'1 2 3; 4 1 6; 7 8 9',
'2 -1 0; -1 2 -1; 0 -1 2',
'-1 -2 3 2; -4 -1 6 2; 7 -8 9 1; 1 -2 1 3',
'1 2 3 4; 5 6 7 8; 9 33 11 12; 13 14 15 17',
'3 1 8 9 6; 6 2 8 10 1; 5 7 2 10 3; 3 2 7 7 9; 3 5 6 1 1',
'-4525/6238  2529/6238 -233/3119 1481/3119 -639/6238;
1033/6238 -1075/6238  342/3119 -447/3119  871/6238;
1299/6238  -289/6238 -204/3119 -390/3119  739/6238;
782/3119  -222/3119  237/3119 -556/3119 -177/3119;
-474/3119   -17/3119  -24/3119  688/3119 -140/3119';

@tests.map: {
my @matrix = .&to-matrix;
say_it( 'Original Matrix:', @matrix );
say_it( 'Gauss-Jordan Inverted Matrix:', gauss-jordan-invert @matrix );
}
```

{{out}}

```Original Matrix:
1  2  3
4  1  6
7  8  9

Gauss-Jordan Inverted Matrix:
-13/16     1/8    3/16
1/8    -1/4     1/8
25/48     1/8   -7/48

Original Matrix:
2  -1   0
-1   2  -1
0  -1   2

Gauss-Jordan Inverted Matrix:
3/4  1/2  1/4
1/2    1  1/2
1/4  1/2  3/4

Original Matrix:
-1  -2   3   2
-4  -1   6   2
7  -8   9   1
1  -2   1   3

Gauss-Jordan Inverted Matrix:
-21/23   17/69  13/138   19/46
-38/23   15/23    1/23   15/23
-16/23   25/69  11/138    9/46
-13/23   16/69   -2/69   13/23

Original Matrix:
1   2   3   4
5   6   7   8
9  33  11  12
13  14  15  17

Gauss-Jordan Inverted Matrix:
19/184  -199/184     -1/46       1/2
1/23     -2/23      1/23         0
-441/184   813/184     -1/46      -3/2
2        -3         0         1

Original Matrix:
3   1   8   9   6
6   2   8  10   1
5   7   2  10   3
3   2   7   7   9
3   5   6   1   1

Gauss-Jordan Inverted Matrix:
-4525/6238   2529/6238   -233/3119   1481/3119   -639/6238
1033/6238  -1075/6238    342/3119   -447/3119    871/6238
1299/6238   -289/6238   -204/3119   -390/3119    739/6238
782/3119   -222/3119    237/3119   -556/3119   -177/3119
-474/3119    -17/3119    -24/3119    688/3119   -140/3119

Original Matrix:
-4525/6238   2529/6238   -233/3119   1481/3119   -639/6238
1033/6238  -1075/6238    342/3119   -447/3119    871/6238
1299/6238   -289/6238   -204/3119   -390/3119    739/6238
782/3119   -222/3119    237/3119   -556/3119   -177/3119
-474/3119    -17/3119    -24/3119    688/3119   -140/3119

Gauss-Jordan Inverted Matrix:
3   1   8   9   6
6   2   8  10   1
5   7   2  10   3
3   2   7   7   9
3   5   6   1   1
```

## Phix

{{trans|Kotlin}} uses ToReducedRowEchelonForm() from [[Reduced_row_echelon_form#Phix]]

```function inverse(sequence mat)
integer len = length(mat)
sequence aug = repeat(repeat(0,2*len),len)
for i=1 to len do
if length(mat[i])!=len then ?9/0 end if -- "Not a square matrix"
for j=1 to len do
aug[i][j] = mat[i][j]
end for
-- augment by identity matrix to right
aug[i][i + len] = 1
end for
aug = ToReducedRowEchelonForm(aug)
sequence inv = repeat(repeat(0,len),len)
-- remove identity matrix to left
for i=1 to len do
for j=len+1 to 2*len do
inv[i][j-len] = aug[i][j]
end for
end for
return inv
end function

constant test = {{ 2, -1,  0},
{-1,  2, -1},
{ 0, -1,  2}}
pp(inverse(test),{pp_Nest,1})
```

{{out}}

```
{{0.75,0.5,0.25},
{0.5,1,0.5},
{0.25,0.5,0.75}}

```

## PowerShell

```
function gauss-jordan-inv([double[][]]\$a) {
\$n = \$a.count
[double[][]]\$b = 0..(\$n-1) | foreach{[double[]]\$row = @(0) * \$n; \$row[\$_] = 1; ,\$row}
for (\$k = 0; \$k -lt \$n; \$k++) {
\$lmax, \$max = \$k, [Math]::Abs(\$a[\$k][\$k])
for (\$l = \$k+1; \$l -lt \$n; \$l++) {
\$tmp = [Math]::Abs(\$a[\$l][\$k])
if(\$max -lt \$tmp) {
\$max, \$lmax = \$tmp, \$l
}
}
if (\$k -ne \$lmax) {
\$a[\$k], \$a[\$lmax] = \$a[\$lmax], \$a[\$k]
\$b[\$k], \$b[\$lmax] = \$b[\$lmax], \$b[\$k]
}
\$akk = \$a[\$k][\$k]
if (0 -eq \$akk) {throw "Irregular matrix"}
for (\$j = 0; \$j -lt \$n; \$j++) {
\$a[\$k][\$j] /= \$akk
\$b[\$k][\$j] /= \$akk
}
for (\$i = 0; \$i -lt \$n; \$i++){
if (\$i -ne \$k) {
\$aik  = \$a[\$i][\$k]
for (\$j = 0; \$j -lt \$n; \$j++) {
\$a[\$i][\$j] -= \$a[\$k][\$j]*\$aik
\$b[\$i][\$j] -= \$b[\$k][\$j]*\$aik
}
}
}
}
\$b
}
function show(\$a) { \$a | foreach{ "\$_"} }

\$a = @(@(@(1, 2, 3), @(4, 1, 6), @(7, 8, 9)))
\$inva = gauss-jordan-inv \$a
"a ="
show \$a
""
"inv(a) ="
show \$inva
""

\$b = @(@(2, -1, 0), @(-1, 2, -1), @(0, -1, 2))
"b ="
show \$b
""
\$invb = gauss-jordan-inv \$b
"inv(b) ="
show \$invb

```

Output:

```
a =
1 2 3
4 1 6
7 8 9

inv(a) =
-0.8125 0.125 0.1875
0.125 -0.25 0.125
0.520833333333333 0.125 -0.145833333333333

b =
2 -1 0
-1 2 -1
0 -1 2

inv(b) =
0.75 0.5 0.25
0.5 1 0.5
0.25 0.5 0.75

```

## Racket

Using the matrix library that comes with default Racket installation

```#lang racket

(require math/matrix
math/array)

(define (inverse M)
(define dim (square-matrix-size M))
(define MI (matrix-augment (list M (identity-matrix dim))))
(submatrix (matrix-row-echelon MI #t #t) (::) (:: dim  #f)))

(define A (matrix [[1 2 3] [4 1 6] [7 8 9]]))

(inverse A)
(matrix-inverse A)
```

{{out}}

```
(array #[#[-13/16 1/8 3/16] #[1/8 -1/4 1/8] #[25/48 1/8 -7/48]])
(array #[#[-13/16 1/8 3/16] #[1/8 -1/4 1/8] #[25/48 1/8 -7/48]])

```

## REXX

```/* REXX */
Parse Arg seed nn
If seed='' Then
seed=23345
If nn='' Then nn=5
If seed='?' Then Do
Say 'rexx gjmi seed n computes a random matrix with n rows and columns'
Say 'Default is 23345 5'
Exit
End
Numeric Digits 50
Call random 1,2,seed
a=''
Do i=1 To nn**2
a=a random(9)+1
End
n2=words(a)
Do n=2 To n2/2
If n**2=n2 Then
Leave
End
If n>n2/2 Then
Call exit 'Not a square matrix:' a '('n2 'elements).'
det=determinante(a,n)
If det=0 Then
Call exit 'Determinant is 0'
Do j=1 To n
Do i=1 To n
Parse Var A a.i.j a
aa.i.j=a.i.j
End
Do ii=1 To n
z=(ii=j)
iii=ii+n
a.iii.j=z
End
End
Call show 1,'The given matrix'
Do m=1 To n-1
If a.m.m=0 Then Do
Do j=m+1 To n
If a.m.j<>0 Then Leave
End
If j>n Then Do
Say 'No pivot>0 found in column' m
Exit
End
Do i=1 To n*2
temp=a.i.m
a.i.m=a.i.j
a.i.j=temp
End
End
Do j=m+1 To n
If a.m.j<>0 Then Do
jj=m
fact=divide(a.m.m,a.m.j)
Do i=1 To n*2
a.i.j=subtract(multiply(a.i.j,fact),a.i.jj)
End
End
End
Call show 2 m
End
Say 'Lower part has all zeros'
Say ''

Do j=1 To n
If denom(a.j.j)<0 Then Do
Do i=1 To 2*n
a.i.j=subtract(0,a.i.j)
End
End
End
Call show 3

Do m=n To 2 By -1
Do j=1 To m-1
jj=m
fact=divide(a.m.j,a.m.jj)
Do i=1 To n*2
a.i.j=subtract(a.i.j,multiply(a.i.jj,fact))
End
End
Call show 4 m
End
Say 'Upper half has all zeros'
Say ''
Do j=1 To n
If decimal(a.j.j)<>1 Then Do
z=a.j.j
Do i=1 To 2*n
a.i.j=divide(a.i.j,z)
End
End
End
Call show 5
Say 'Main diagonal has all ones'
Say ''

Do j=1 To n
Do i=1 To n
z=i+n
a.i.j=a.z.j
End
End
Call show 6,'The inverse matrix'

do i = 1 to n
do j = 1 to n
sum=0
Do k=1 To n
End
c.i.j = sum
end
End
Call showc 7,'The product of input and inverse matrix'
Exit

show:
Parse Arg num,text
Say 'show' arg(1) text
If arg(1)<>6 Then rows=n*2
Else rows=n
len=0
Do j=1 To n
Do i=1 To rows
len=max(len,length(a.i.j))
End
End
Do j=1 To n
ol=''
Do i=1 To rows
ol=ol||right(a.i.j,len+1)
End
Say ol
End
Say ''
Return

showc:
Parse Arg num,text
Say text
clen=0
Do j=1 To n
Do i=1 To n
clen=max(clen,length(c.i.j))
End
End
Do j=1 To n
ol=''
Do i=1 To n
ol=ol||right(c.i.j,clen+1)
End
Say ol
End
Say ''
Return

denom: Procedure
/* Return the denominator */
Parse Arg d '/' n
Return d

decimal: Procedure
/* compute the fraction's value */
Parse Arg a
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an

gcd: procedure
/**********************************************************************
* Greatest commn divisor
**********************************************************************/
Parse Arg a,b
If b = 0 Then Return abs(a)
Return gcd(b,a//b)

Parse Arg a,b
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
If pos('/',b)=0 Then b=b'/1'; Parse Var b bd '/' bn
Return sum

multiply: Procedure
Parse Arg a,b
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
If pos('/',b)=0 Then b=b'/1'; Parse Var b bd '/' bn
Return prd

subtract: Procedure
Parse Arg a,b
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
If pos('/',b)=0 Then b=b'/1'; Parse Var b bd '/' bn
Return div

divide: Procedure
Parse Arg a,b
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
If pos('/',b)=0 Then b=b'/1'; Parse Var b bd '/' bn
sn=an*bd
g=gcd(sd,sn)
Select
When sd=0 Then res='0'
When abs(sn/g)=1 Then res=(sd/g)*sign(sn/g)
Otherwise Do
den=sd/g
nom=sn/g
If nom<0 Then Do
If den<0 Then den=abs(den)
Else den=-den
nom=abs(nom)
End
res=den'/'nom
End
End
Return res

determinante: Procedure
/* REXX ***************************************************************
* determinant.rex
* compute the determinant of the given square matrix
* Input: as: the representation of the matrix as vector (n**2 elements)
* 21.05.2013 Walter Pachl
**********************************************************************/
Parse Arg as,n
Do i=1 To n
Do j=1 To n
Parse Var as a.i.j as
End
End
Select
When n=2 Then det=subtract(multiply(a.1.1,a.2.2),multiply(a.1.2,a.2.1))
When n=3 Then Do
det=multiply(multiply(a.1.1,a.2.2),a.3.3)
det=subtract(det,multiply(multiply(a.1.3,a.2.2),a.3.1))
det=subtract(det,multiply(multiply(a.1.2,a.2.1),a.3.3))
det=subtract(det,multiply(multiply(a.1.1,a.2.3),a.3.2))
End
Otherwise Do
det=0
Do k=1 To n
sign=((-1)**(k+1))
If sign=1 Then
Else
det=subtract(det,multiply(a.1.k,determinante(subm(k),n-1)))
End
End
End
Return det

subm: Procedure Expose a. n
/**********************************************************************
* compute the submatrix resulting when row 1 and column k are removed
* Input: a.*.*, k
* Output: bs the representation of the submatrix as vector
**********************************************************************/
Parse Arg k
bs=''
do i=2 To n
Do j=1 To n
If j=k Then Iterate
bs=bs a.i.j
End
End
Return bs

Exit: Say arg(1)
```

{{out}} Using the defaults for seed and n

```show 1 The given matrix
10  3  8  6  3  1  0  0  0  0
5  7  8  8  2  0  1  0  0  0
4 10  5  4  7  0  0  1  0  0
9  4  5  3  3  0  0  0  1  0
6  3  3  3  7  0  0  0  0  1

show 2 1
10     3     8     6     3     1     0     0     0     0
0    11     8    10     1    -1     2     0     0     0
0    22   9/2     4  29/2    -1     0   5/2     0     0
0  13/9 -22/9  -8/3   1/3    -1     0     0  10/9     0
0     2    -3    -1  26/3    -1     0     0     0   5/3

show 2 2
10       3       8       6       3       1       0       0       0       0
0      11       8      10       1      -1       2       0       0       0
0       0   -23/4      -8    25/4     1/2      -2     5/4       0       0
0       0 -346/13 -394/13   20/13  -86/13      -2       0  110/13       0
0       0   -49/2   -31/2   140/3    -9/2      -2       0       0    55/6

show 2 3
10         3         8         6         3         1         0         0         0         0
0        11         8        10         1        -1         2         0         0         0
0         0     -23/4        -8      25/4       1/2        -2       5/4         0         0
0         0         0  1005/692 -4095/692 -1335/692  1085/692      -5/4  1265/692         0
0         0         0   855/196    395/84  -305/196     75/49      -5/4         0  1265/588

show 2 4
10            3            8            6            3            1            0            0            0            0
0           11            8           10            1           -1            2            0            0            0
0            0        -23/4           -8         25/4          1/2           -2          5/4            0            0
0            0            0     1005/692    -4095/692    -1335/692     1085/692         -5/4     1265/692            0
0            0            0            0 221375/29583   13915/9861 -13915/13148  16445/19722    -1265/692 84755/118332

Lower part has all zeros

show 3
10            3            8            6            3            1            0            0            0            0
0           11            8           10            1           -1            2            0            0            0
0            0         23/4            8        -25/4         -1/2            2         -5/4            0            0
0            0            0     1005/692    -4095/692    -1335/692     1085/692         -5/4     1265/692            0
0            0            0            0 221375/29583   13915/9861 -13915/13148  16445/19722    -1265/692 84755/118332

show 4 5
10            3            8            6            0       76/175      297/700     -117/350      513/700     -201/700
0           11            8           10            0     -208/175     1499/700      -39/350      171/700      -67/700
0            0         23/4            8            0        19/28      125/112       -31/56     -171/112       67/112
0            0            0     1005/692            0   -1407/1730  10117/13840   -4087/6920   5293/13840   7839/13840
0            0            0            0 221375/29583   13915/9861 -13915/13148  16445/19722    -1265/692 84755/118332

show 4 4
10            3            8            0            0      664/175    -1817/700      737/350     -593/700    -1839/700
0           11            8            0            0      772/175   -6073/2100    4153/1050   -5017/2100    -2797/700
0            0         23/4            0            0     3611/700  -24449/8400   11339/4200  -30521/8400   -7061/2800
0            0            0     1005/692            0   -1407/1730  10117/13840   -4087/6920   5293/13840   7839/13840
0            0            0            0 221375/29583   13915/9861 -13915/13148  16445/19722    -1265/692 84755/118332

show 4 3
10            3            0            0            0     -592/175    3053/2100   -1733/1050    8837/2100      617/700
0           11            0            0            0     -484/175    2431/2100     209/1050    5599/2100     -341/700
0            0         23/4            0            0     3611/700  -24449/8400   11339/4200  -30521/8400   -7061/2800
0            0            0     1005/692            0   -1407/1730  10117/13840   -4087/6920   5293/13840   7839/13840
0            0            0            0 221375/29583   13915/9861 -13915/13148  16445/19722    -1265/692 84755/118332

show 4 2
10            0            0            0            0       -92/35      239/210     -179/105      731/210        71/70
0           11            0            0            0     -484/175    2431/2100     209/1050    5599/2100     -341/700
0            0         23/4            0            0     3611/700  -24449/8400   11339/4200  -30521/8400   -7061/2800
0            0            0     1005/692            0   -1407/1730  10117/13840   -4087/6920   5293/13840   7839/13840
0            0            0            0 221375/29583   13915/9861 -13915/13148  16445/19722    -1265/692 84755/118332

Upper half has all zeros

show 5
1          0          0          0          0    -46/175   239/2100  -179/1050   731/2100     71/700
0          1          0          0          0    -44/175   221/2100    19/1050   509/2100    -31/700
0          0          1          0          0    157/175 -1063/2100   493/1050 -1327/2100   -307/700
0          0          0          1          0     -14/25    151/300    -61/150     79/300     39/100
0          0          0          0          1     33/175    -99/700     39/350   -171/700     67/700

Main diagonal has all ones

show 6 The inverse matrix
-46/175   239/2100  -179/1050   731/2100     71/700
-44/175   221/2100    19/1050   509/2100    -31/700
157/175 -1063/2100   493/1050 -1327/2100   -307/700
-14/25    151/300    -61/150     79/300     39/100
33/175    -99/700     39/350   -171/700     67/700

The product of input and inverse matrix
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
```

## Sidef

Uses the '''rref(M)''' function from [https://rosettacode.org/wiki/Reduced_row_echelon_form#Sidef Reduced row echelon form]. {{trans|Perl 6}}

```func gauss_jordan_invert (M) {

var I = M.len.of {|i|
M.len.of {|j|
i == j ? 1 : 0
}
}

var A = gather {
^M -> each {|i| take(M[i] + I[i]) }
}

rref(A).map { .last(M.len) }
}

var A = [
[-1, -2, 3, 2],
[-4, -1, 6, 2],
[ 7, -8, 9, 1],
[ 1, -2, 1, 3],
]

say gauss_jordan_invert(A).map {
.map { "%6s" % .as_rat }.join("  ")
}.join("\n")
```

{{out}}

```
-21/23   17/69  13/138   19/46
-38/23   15/23    1/23   15/23
-16/23   25/69  11/138    9/46
-13/23   16/69   -2/69   13/23

```

## VBA

{{trans|Phix}} Uses ToReducedRowEchelonForm() from [[Reduced_row_echelon_form#VBA]]

```Private Function inverse(mat As Variant) As Variant
Dim len_ As Integer: len_ = UBound(mat)
Dim tmp() As Variant
ReDim tmp(2 * len_ + 1)
Dim aug As Variant
ReDim aug(len_)
For i = 0 To len_
If UBound(mat(i)) <> len_ Then Debug.Print 9 / 0 '-- "Not a square matrix"
aug(i) = tmp
For j = 0 To len_
aug(i)(j) = mat(i)(j)
Next j
'-- augment by identity matrix to right
aug(i)(i + len_ + 1) = 1
Next i
aug = ToReducedRowEchelonForm(aug)
Dim inv As Variant
inv = mat
'-- remove identity matrix to left
For i = 0 To len_
For j = len_ + 1 To 2 * len_ + 1
inv(i)(j - len_ - 1) = aug(i)(j)
Next j
Next i
inverse = inv
End Function

Public Sub main()
Dim test As Variant
test = inverse(Array( _
Array(2, -1, 0), _
Array(-1, 2, -1), _
Array(0, -1, 2)))
For i = LBound(test) To UBound(test)
For j = LBound(test(0)) To UBound(test(0))
Debug.Print test(i)(j),
Next j
Debug.Print
Next i
End Sub
```

{{out}}

``` 0,75          0,5           0,25
0,5           1             0,5
0,25          0,5           0,75
```

## zkl

This uses GSL to invert a matrix via LU decomposition, not Gauss-Jordan.

```var [const] GSL=Import.lib("zklGSL");    // libGSL (GNU Scientific Library)
m:=GSL.Matrix(3,3).set(1,2,3, 4,1,6, 7,8,9);
i:=m.invert();
i.format(10,4).println("\n");
(m*i).format(10,4).println();
```

{{out}}

```
-0.8125,    0.1250,    0.1875
0.1250,   -0.2500,    0.1250
0.5208,    0.1250,   -0.1458

1.0000,    0.0000,    0.0000
-0.0000,    1.0000,    0.0000
-0.0000,    0.0000,    1.0000

```
```m:=GSL.Matrix(3,3).set(2,-1,0, -1,2,-1, 0,-1,2);
m.invert().format(10,4).println("\n");
```

{{out}}

```
0.7500,    0.5000,    0.2500
0.5000,    1.0000,    0.5000
0.2500,    0.5000,    0.7500

```