Adjoint and Inverse of a Matrix

Given a square matrix, find adjoint and inverse of the matrix.

We strongly recommend you to refer below as a prerequisite of this.
Determinant of a Matrix

What is Adjoint?
Adjoint (or Adjugate) of a matrix is the matrix obtained by taking transpose of the cofactor matrix of a given square matrix is called its Adjoint or Adjugate matrix. The Adjoint of any square matrix ‘A’ (say) is represented as Adj(A).



Example:

Below example and explanation are taken from here.
5  -2  2  7
1   0  0  3
-3  1  5  0
3  -1 -9  4

For instance, the cofactor of the top left corner '5' is
 + |0   0   3|
...|1   5   0| = 3(1 * -9 - (-1) * 5) = -12.
...|-1 -9   4|
(The minor matrix is formed by deleting the row 
 and column of the given entry.)

As another sample, the cofactor of the top row corner '-2' is
  -|1   0  3|
...|-3  5  0| = - [1 (20 - 0) - 0 + 3 (27 - 15)] = -56.
...|3  -9  4|

Proceeding like this, we obtain the matrix
[-12  -56   4   4]
[76   208   4   4]
[-60  -82  -2  20]
[-36  -58  -10 12]

Finally, to get the adjoint, just take the previous
matrix's transpose:
[-12   76 -60  -36]
[-56  208 -82  -58]
[4     4   -2  -10]
[4     4   20   12] 

Important properties:

  • Product of a square matrix A with its adjoint yields a diagonal matrix, where each diagonal entry is equal to determinant of A.
    i.e.,

    A.adj(A) = det(A).I 
    
    I  => Identity matrix of same order as of A.
    det(A) => Determinant value of A 
  • A non zero square matrix ‘A’ of order n is said to be invertible if there exists a unique square matrix ‘B’ of order n such that,
       A.B = B.A = I
    The matrix 'B' is said to be inverse of 'A'.
    i.e.,  B = A-1

How to find Adjoint?
We follow definition given above.


Let A[N][N] be input matrix.

1) Create a matrix adj[N][N] store the adjoint matrix.
2) For every entry A[i][j] in input matrix where 0 <= i < N
   and 0 <= j < N.
    a) Find cofactor of A[i][j]
    b) Find sign of entry.  Sign is + if (i+j) is even else
       sign is odd.
    c) Place the cofactor at adj[j][i]

How to find Inverse?
Inverse of a matrix exists only if the matrix is non-singular i.e., determinant should not be 0.
Using determinant and adjoint, we can easily find the inverse of a square matrix using below formula,

  If det(A) != 0
    A-1 = adj(A)/det(A)
  Else
    "Inverse doesn't exist"  

Inverse is used to find the solution to a system of linear equation.

Below is C++ implementation for finding adjoint and inverse of a matrix.

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// C++ program to find adjoint and inverse of a matrix
#include<bits/stdc++.h>
using namespace std;
#define N 4
  
// Function to get cofactor of A[p][q] in temp[][]. n is current
// dimension of A[][]
void getCofactor(int A[N][N], int temp[N][N], int p, int q, int n)
{
    int i = 0, j = 0;
  
    // Looping for each element of the matrix
    for (int row = 0; row < n; row++)
    {
        for (int col = 0; col < n; col++)
        {
            //  Copying into temporary matrix only those element
            //  which are not in given row and column
            if (row != p && col != q)
            {
                temp[i][j++] = A[row][col];
  
                // Row is filled, so increase row index and
                // reset col index
                if (j == n - 1)
                {
                    j = 0;
                    i++;
                }
            }
        }
    }
}
  
/* Recursive function for finding determinant of matrix.
   n is current dimension of A[][]. */
int determinant(int A[N][N], int n)
{
    int D = 0; // Initialize result
  
    //  Base case : if matrix contains single element
    if (n == 1)
        return A[0][0];
  
    int temp[N][N]; // To store cofactors
  
    int sign = 1;  // To store sign multiplier
  
     // Iterate for each element of first row
    for (int f = 0; f < n; f++)
    {
        // Getting Cofactor of A[0][f]
        getCofactor(A, temp, 0, f, n);
        D += sign * A[0][f] * determinant(temp, n - 1);
  
        // terms are to be added with alternate sign
        sign = -sign;
    }
  
    return D;
}
  
// Function to get adjoint of A[N][N] in adj[N][N].
void adjoint(int A[N][N],int adj[N][N])
{
    if (N == 1)
    {
        adj[0][0] = 1;
        return;
    }
  
    // temp is used to store cofactors of A[][]
    int sign = 1, temp[N][N];
  
    for (int i=0; i<N; i++)
    {
        for (int j=0; j<N; j++)
        {
            // Get cofactor of A[i][j]
            getCofactor(A, temp, i, j, N);
  
            // sign of adj[j][i] positive if sum of row
            // and column indexes is even.
            sign = ((i+j)%2==0)? 1: -1;
  
            // Interchanging rows and columns to get the
            // transpose of the cofactor matrix
            adj[j][i] = (sign)*(determinant(temp, N-1));
        }
    }
}
  
// Function to calculate and store inverse, returns false if
// matrix is singular
bool inverse(int A[N][N], float inverse[N][N])
{
    // Find determinant of A[][]
    int det = determinant(A, N);
    if (det == 0)
    {
        cout << "Singular matrix, can't find its inverse";
        return false;
    }
  
    // Find adjoint
    int adj[N][N];
    adjoint(A, adj);
  
    // Find Inverse using formula "inverse(A) = adj(A)/det(A)"
    for (int i=0; i<N; i++)
        for (int j=0; j<N; j++)
            inverse[i][j] = adj[i][j]/float(det);
  
    return true;
}
  
// Generic function to display the matrix.  We use it to display
// both adjoin and inverse. adjoin is integer matrix and inverse
// is a float.
template<class T>
void display(T A[N][N])
{
    for (int i=0; i<N; i++)
    {
        for (int j=0; j<N; j++)
            cout << A[i][j] << " ";
        cout << endl;
    }
}
  
// Driver program
int main()
{
    int A[N][N] = { {5, -2, 2, 7},
                    {1, 0, 0, 3},
                    {-3, 1, 5, 0},
                    {3, -1, -9, 4}};
  
    int adj[N][N];  // To store adjoint of A[][]
  
    float inv[N][N]; // To store inverse of A[][]
  
    cout << "Input matrix is :\n";
    display(A);
  
    cout << "\nThe Adjoint is :\n";
    adjoint(A, adj);
    display(adj);
  
    cout << "\nThe Inverse is :\n";
    if (inverse(A, inv))
        display(inv);
  
    return 0;
}

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Output:

The Adjoint is :
-12 76 -60 -36 
-56 208 -82 -58 
4 4 -2 -10 
4 4 20 12 

The Inverse is :
-0.136364 0.863636 -0.681818 -0.409091 
-0.636364 2.36364 -0.931818 -0.659091 
0.0454545 0.0454545 -0.0227273 -0.113636 
0.0454545 0.0454545 0.227273 0.136364

Please refer https://www.geeksforgeeks.org/determinant-of-a-matrix/ for details of getCofactor() and determinant().

References:
https://www.geeksforgeeks.org/determinant-of-a-matrix/
https://en.wikipedia.org/wiki/Adjugate_matrix

This article is contributed by Ashutosh Kumar. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above



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