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.

// 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; 
} 

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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