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Finding inverse of a matrix using Gauss Jordan Method in C++
In this problem, we are given an 2D matrix mat[][]. Our task is to find inverse of a matrix using Gauss Jordan Method.
Now, lets understand the basics of the problem,
MATRIX is a two dimensional array of numbers.
Example
$\begin{bmatrix}2&5&4 \1&6&7 \9&3&8\end{bmatrix}$
Inverse of Matrix [A-1] −
It is an operation performed on square matrix. The following are the properties that are required for a matrix to have an inverse −
Initial matrix should be square matrix.
It must be non-singular matrix.
An identity matrix I exist for the matrix A such that,
$$AA^{-1} = A^{-1}.A = I$$
Their is a formula that can be used to find the inverse of ant given matrix. It is
$A^{-1}\:=\:\left(\frac{adj(A)}{\det(A)}\right)$
adj(A) is adjoint of matrix A
det(A) is determinant of matrix A.
Their are multiple ways using with we can find the inverse of a matrix. In this article, we will be learning about Gauss Jordan Method which is also known as Elementary Row Opeation.
It is a step by step method to find the inverse of a matrix, Here are the steps involved −
Finding augmented matrix using the identity matrix.
Find the echelon form of the matrix by performing row reduction operation on the augmented matrix found in step 1.
Some operation that can be performed on the augmented matrix in the process are
Row interchange (you can interchange any two rows)
Multiplication (each elements of the row can be multiplied by a constant values other than 0).
Row Interchange (replace the row with the sum of the row and constant multiple of another row of matrix).
Example
Program to illustrate the working of our solution
#include <iostream>
#include <vector>
using namespace std;
void printMatrixValues(float** arr, int n, int m){
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
cout<<arr[i][j]<<"\t";
}
cout<<endl;
}
return;
}
void printInverseMatrix(float** arr, int n, int m){
for (int i = 0; i < n; i++) {
for (int j = n; j < m; j++) {
printf("%.3f\t", arr[i][j]);
}
cout<<endl;
}
return;
}
void findInvMatGaussJordan(float** mat, int order){
float temp;
printf("The inverse of matrix : A = \n");
printMatrixValues(mat, order, order);
for (int i = 0; i < order; i++) {
for (int j = 0; j < 2 * order; j++) {
if (j == (i + order))
mat[i][j] = 1;
}
}
for (int i = order - 1; i > 0; i--) {
if (mat[i - 1][0] < mat[i][0]) {
float* temp = mat[i];
mat[i] = mat[i - 1];
mat[i - 1] = temp;
}
}
for (int i = 0; i < order; i++) {
for (int j = 0; j < order; j++) {
if (j != i) {
temp = mat[j][i] / mat[i][i];
for (int k = 0; k < 2 * order; k++) {
mat[j][k] -= mat[i][k] * temp;
}
}
}
}
for (int i = 0; i < order; i++) {
temp = mat[i][i];
for (int j = 0; j < 2 * order; j++) {
mat[i][j] = mat[i][j] / temp;
}
}
cout<<"A' =\n";
printInverseMatrix(mat, order, 2 * order);
return;
}
int main(){
int order = 3;
float** mat = new float*[20];
for (int i = 0; i < 20; i++)
mat[i] = new float[20];
mat[0][0] = 6; mat[0][1] = 9; mat[0][2] = 5;
mat[1][0] = 8; mat[1][1] = 3; mat[1][2] = 2;
mat[2][0] = 1; mat[2][1] = 4; mat[2][2] = 7;
findInvMatGaussJordan(mat, order);
return 0;
}Output
The inverse of matrix : A = 6 9 5 8 3 2 1 4 7 A' = -0.049 0.163 -0.011 0.205 -0.141 -0.106 -0.110 0.057 0.205
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