Adjoint of n x n Matrix in C++

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Adjoint of a Matrix is the transpose of the co-factors of the given matrix. It is the core Mathematics concept used in calculating inverse of a matrix. In this Article, I am going to share with you the code of adjoint of n x n matrix in C++ where n=1,2 or 3.

What is Adjoint of n x n Matrix

The adjoint of an n x n matrix is known as the adjugate or classical adjoint. It is a matrix that is related to the original matrix and is used in various mathematical operations, especially in finding the inverse of a matrix. The adjoint is denoted as “adj(A)” for a matrix A.

To find the adjoint of an n x n matrix A, you need to follow these steps:

  1. Calculate the cofactor matrix of A.
  • For each element A[i][j] of A, calculate the cofactor C[i][j]. The cofactor of an element is the determinant of the matrix obtained by removing the i-th row and j-th column from A, multiplied by (-1)^(i+j). In mathematical notation:
    C[i][j] = (-1)^(i+j) * det(M[i][j]),
    where M[i][j] is the (n-1) x (n-1) matrix obtained by removing the i-th row and j-th column from A.
  1. Transpose the cofactor matrix.
  • Swap the rows and columns of the cofactor matrix obtained in step 1 to get the adjoint matrix.

Mathematically, if C is the cofactor matrix of A, then the adjoint matrix adj(A) is given by:
adj(A) = transpose(C)

The adjoint of a matrix plays a crucial role in finding the inverse of a matrix. Specifically, if A is an invertible (non-singular) matrix, then you can find the inverse of A as follows:
A^(-1) = (1 / det(A)) * adj(A)
where det(A) is the determinant of A.

Keep in mind that the adjoint is only defined for square matrices (n x n matrices), and it’s used primarily in linear algebra and matrix operations.

Code in C++

//ADJOINT of matrices :)

#include<iostream>
#include<math.h>
#include<conio.h>
using namespace std;
int main()
{
	int n;
	system("color b4 ");
	do
	{
	cout<<"\n\n\tEnter Size of Matrix : \n\n";
	cout<<"\t1-One x One "<<endl;
	cout<<"\t2-Two x Two"<<endl;
	cout<<"\t3-Three x Three "<<endl<<"\t";
	cin>>n;
	}
	while(n>4 || n<1) ;
	//system("CLS");
	//system("color F3");
	int mat[n][n],adj[n][n];
	cout<<"\n\n\tEnter Elements now \n";
	int i,j;
	//enter elements
	for(i=0;i<n;i++)
	{
		for(j=0;j<n;j++)
		{
			cout<<"\tElement ["<<i+1<<"]["<<j+1<<"] : ";cin>>mat[i][j];
		}
	}
	cout<<"\n\t";
	if(n==1)
	{
		adj[0][0]=mat[0][0];
		
	}
	//2x2 matrix determinent
	if(n==2)
	{
		//2X2 matrix determinent
		adj[0][0]=mat[1][1];
		adj[0][1]=-1*mat[0][1];
		adj[1][0]=-1*mat[1][0];
		adj[1][1]=mat[0][0];
	}
	if(n==3)
	{
		//3X3 matrix adjoint
		//cofactor 11
		adj[0][0]=pow(-1,1+1)*(mat[1][1]*mat[2][2]-mat[1][2]*mat[2][1]);
		//cofactor 12
		adj[0][1]=pow(-1,1+2)*(mat[1][0]*mat[2][2]-mat[1][2]*mat[2][0]);
		//cofactor 13
		adj[0][2]=pow(-1,1+3)*(mat[1][0]*mat[2][1]-mat[1][1]*mat[2][0]);
		//cofactor 21
		adj[1][0]=pow(-1,2+1)*(mat[0][1]*mat[2][2]-mat[0][2]*mat[2][1]);
		//cofactor 22
		adj[1][1]=pow(-1,2+2)*(mat[0][0]*mat[2][2]-mat[0][2]*mat[2][0]);
		//cofactor 23
		adj[1][2]=pow(-1,2+3)*(mat[0][0]*mat[2][1]-mat[0][1]*mat[2][0]);
		//cofactor 31
		adj[2][0]=pow(-1,3+1)*(mat[0][1]*mat[1][2]-mat[0][2]*mat[1][1]);
		//cofactor 32
		adj[2][1]=pow(-1,3+2)*(mat[0][0]*mat[1][2]-mat[0][2]*mat[1][0]);
		//cofactor 33
		adj[2][2]=pow(-1,3+3)*(mat[0][0]*mat[1][1]-mat[0][1]*mat[1][0]);
	}
	//print matrix adjoint 
	cout<<"\n\tADJOINT OF "<<n<<" x "<<n<<" MATRIX IS  \n\n";
	for(i=0;i<n;i++)
	{
		for(j=0;j<n;j++)
		{
			cout<<"\t"<<adj[j][i]<<"\t";
		}
		cout<<endl;
	}
	getch();
}

Output

Adjoint of n x n matrix in C++
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