Adjoint and Inverse of a Matrix

Last Updated : 01 Nov, 2024

Given a square matrix, find the adjoint and inverse of the matrix. We strongly recommend you to refer determinant of matrix as a prerequisite for this.

Adjoint (or Adjugate) of a matrix is the matrix obtained by taking the 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:

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

adj(AB) = (adj B).(adj A)
adj( k A) = k^n-1 adj(A)
A^-1 = (adj A) / |A|
(A^-1)^-1 = A
(AB)^-1 = B^-1A^-1

How to find Adjoint?

We follow the 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 the 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 equations.

Below are implementations for finding adjoint and inverse of a matrix.

C++

#include <iostream> #include <vector> using namespace std;  // Function to get cofactor of mat[p][q] in cof[][]. n is // current dimension of mat[][] void getCof(vector<vector<int>>& mat, vector<vector<int>>& cof,                                          int p, int q, int n) {     int i = 0, j = 0;     for (int row = 0; row < n; row++) {         for (int col = 0; col < n; col++) {             if (row != p && col != q) {                 cof[i][j++] = mat[row][col];                 if (j == n - 1) {                     j = 0;                     i++;                 }             }         }     } }  // Recursive function for finding determinant  // of matrix mat of dimension n int getDet(vector<vector<int>>& mat, int n) {     if (n == 1) return mat[0][0];          int det = 0;        // To store cofactors     vector<vector<int>> cof(mat.size(), vector<int>(mat.size()));        int sign = 1;     for (int f = 0; f < n; f++) {         getCof(mat, cof, 0, f, n);         det += sign * mat[0][f] * getDet(cof, n - 1);         sign = -sign;     }     return det; }  // Function to get adjoint of mat in adj void adjoint(vector<vector<int>>& mat, vector<vector<int>>& adj) {     int n = mat.size();     if (n == 1) {         adj[0][0] = 1;         return;     }          int sign = 1;     vector<vector<int>> cof(n, vector<int>(n));     for (int i = 0; i < n; i++) {         for (int j = 0; j < n; j++) {             getCof(mat, cof, i, j, n);             sign = ((i + j) % 2 == 0) ? 1 : -1;             adj[j][i] = sign * getDet(cof, n - 1);         }     } }  // Function to calculate and store inverse, returns  // false if matrix is singular bool inverse(vector<vector<int>>& mat, vector<vector<float>>& inv) {     int n = mat.size();     int det = getDet(mat, n);     if (det == 0) {         cout << "Singular matrix, can't find its inverse";         return false;     }      vector<vector<int>> adj(n, vector<int>(n));     adjoint(mat, adj);      for (int i = 0; i < n; i++)         for (int j = 0; j < n; j++)             inv[i][j] = adj[i][j] / float(det);      return true; }  int main() {     vector<vector<int>> mat = { { 5, -2, 2, 7 },                                 { 1, 0, 0, 3 },                                 { -3, 1, 5, 0 },                                 { 3, -1, -9, 4 } };      int n = mat.size();     vector<vector<int>> adj(n, vector<int>(n)); // To store adjoint     vector<vector<float>> inv(n, vector<float>(n)); // To store inverse      // Print the input matrix     cout << "Input matrix is:\n";     for (auto& row : mat) {         for (int val : row) cout << val << " ";         cout << endl;     }      // Print the adjoint matrix     cout << "\nThe Adjoint is:\n";     adjoint(mat, adj);     for (auto& row : adj) {         for (int val : row) cout << val << " ";         cout << endl;     }      // Print the inverse matrix if it exists     cout << "\nThe Inverse is:\n";     if (inverse(mat, inv)) {         for (auto& row : inv) {             for (float val : row) cout << val << " ";             cout << endl;         }     }      return 0; }

#include <stdio.h>  #define N 4  // Function to get cofactor of mat[p][q] in cof[][]. n is // current dimension of mat[][] void getCof(int mat[N][N], int cof[N][N], int p, int q, int n) {     int i = 0, j = 0;     for (int row = 0; row < n; row++) {         for (int col = 0; col < n; col++) {             if (row != p && col != q) {                 cof[i][j++] = mat[row][col];                 if (j == n - 1) {                     j = 0;                     i++;                 }             }         }     } }  // Recursive function for finding determinant of matrix mat of dimension n int getDet(int mat[N][N], int n) {     if (n == 1) return mat[0][0];     int det = 0;          int cof[N][N];     int sign = 1;     for (int f = 0; f < n; f++) {         getCof(mat, cof, 0, f, n);         det += sign * mat[0][f] * getDet(cof, n - 1);         sign = -sign;     }     return det; }  // Function to get adjoint of mat in adj void adjoint(int mat[N][N], double adj[N][N]) {     if (N == 1) {         adj[0][0] = 1;         return;     }          int sign = 1;     int cof[N][N];     for (int i = 0; i < N; i++) {         for (int j = 0; j < N; j++) {             getCof(mat, cof, i, j, N);             sign = ((i + j) % 2 == 0) ? 1 : -1;             adj[j][i] = sign * getDet(cof, N - 1);         }     } }  // Function to calculate and store inverse, returns 0 if matrix is singular int inverse(int mat[N][N], double inv[N][N]) {     int det = getDet(mat, N);     if (det == 0) {         printf("Singular matrix, can't find its inverse\n");         return 0;     }      double adj[N][N];     adjoint(mat, adj);      for (int i = 0; i < N; i++)         for (int j = 0; j < N; j++)             inv[i][j] = adj[i][j] / det;      return 1; }  int main() {     int mat[N][N] = { { 5, -2, 2, 7 },                       { 1, 0, 0, 3 },                       { -3, 1, 5, 0 },                       { 3, -1, -9, 4 } };      double adj[N][N];     double inv[N][N];      // Print the input matrix     printf("Input matrix is:\n");     for (int i = 0; i < N; i++) {         for (int j = 0; j < N; j++) {             printf("%d ", mat[i][j]);         }         printf("\n");     }      // Print the adjoint matrix     printf("\nThe Adjoint is:\n");     adjoint(mat, adj);     for (int i = 0; i < N; i++) {         for (int j = 0; j < N; j++) {             printf("%lf ", adj[i][j]);         }         printf("\n");     }      // Print the inverse matrix if it exists     printf("\nThe Inverse is:\n");     if (inverse(mat, inv)) {         for (int i = 0; i < N; i++) {             for (int j = 0; j < N; j++) {                 printf("%lf ", inv[i][j]);             }             printf("\n");         }     }     return 0; }

Java

// Importing required classes import java.util.Arrays; import java.util.Scanner;  public class GfG {        // Function to get cofactor of mat[p][q] in cof[][]     static void getCof(int[][] mat, int[][] cof, int p, int q, int n) {         int i = 0, j = 0;         for (int row = 0; row < n; row++) {             for (int col = 0; col < n; col++) {                 if (row != p && col != q) {                     cof[i][j++] = mat[row][col];                     if (j == n - 1) {                         j = 0;                         i++;                     }                 }             }         }     }      // Recursive function for finding determinant of     // matrix mat of dimension n     static int getDet(int[][] mat, int n) {         if (n == 1) return mat[0][0];         int det = 0;         int[][] cof = new int[n][n];         int sign = 1;         for (int f = 0; f < n; f++) {             getCof(mat, cof, 0, f, n);             det += sign * mat[0][f] * getDet(cof, n - 1);             sign = -sign;         }         return det;     }      // Function to get adjoint of mat in adj     static void adjoint(int[][] mat, int[][] adj) {         int n = mat.length;         if (n == 1) {             adj[0][0] = 1;             return;         }         int sign;         int[][] cof = new int[n][n];         for (int i = 0; i < n; i++) {             for (int j = 0; j < n; j++) {                 getCof(mat, cof, i, j, n);                 sign = ((i + j) % 2 == 0) ? 1 : -1;                 adj[j][i] = sign * getDet(cof, n - 1);             }         }     }      // Function to calculate and store inverse, returns      // false if matrix is singular     static boolean inverse(int[][] mat, float[][] inv) {         int n = mat.length;         int det = getDet(mat, n);         if (det == 0) {             System.out.println("Singular matrix, can't find its inverse");             return false;         }         int[][] adj = new int[n][n];         adjoint(mat, adj);         for (int i = 0; i < n; i++)             for (int j = 0; j < n; j++)                 inv[i][j] = (float) adj[i][j] / det;         return true;     }      public static void main(String[] args) {         int[][] mat = { { 5, -2, 2, 7 }, { 1, 0, 0, 3 },                          { -3, 1, 5, 0 }, { 3, -1, -9, 4 } };         int n = mat.length;         int[][] adj = new int[n][n]; // To store adjoint         float[][] inv = new float[n][n]; // To store inverse          // Print the input matrix         System.out.println("Input matrix is:");         for (int[] row : mat) {             System.out.println(Arrays.toString(row));         }          // Print the adjoint matrix         System.out.println("\nThe Adjoint is:");         adjoint(mat, adj);         for (int[] row : adj) {             System.out.println(Arrays.toString(row));         }          // Print the inverse matrix if it exists         System.out.println("\nThe Inverse is:");         if (inverse(mat, inv)) {             for (float[] row : inv) {                 System.out.println(Arrays.toString(row));             }         }     } }

Python

# Function to get cofactor of mat[p][q] in cof[][] def get_cof(mat, cof, p, q, n):     i = 0     j = 0     for row in range(n):         for col in range(n):             if row != p and col != q:                 cof[i][j] = mat[row][col]                 j += 1                 if j == n - 1:                     j = 0                     i += 1  # Recursive function for finding determinant # of matrix mat of dimension n def get_det(mat, n):     if n == 1:         return mat[0][0]     det = 0     cof = [[0] * n for _ in range(n)]  # To store cofactors     sign = 1     for f in range(n):         get_cof(mat, cof, 0, f, n)         det += sign * mat[0][f] * get_det(cof, n - 1)         sign = -sign     return det  # Function to get adjoint of mat in adj def adjoint(mat, adj):     n = len(mat)     if n == 1:         adj[0][0] = 1         return     sign = 1     cof = [[0] * n for _ in range(n)]     for i in range(n):         for j in range(n):             get_cof(mat, cof, i, j, n)             sign = 1 if (i + j) % 2 == 0 else -1             adj[j][i] = sign * get_det(cof, n - 1)  # Function to calculate and store inverse, returns  # false if matrix is singular def inverse(mat):     n = len(mat)     det = get_det(mat, n)     if det == 0:         print("Singular matrix, can't find its inverse")         return None     adj = [[0] * n for _ in range(n)]     adjoint(mat, adj)     inv = [[adj[i][j] / det for j in range(n)] for i in range(n)]     return inv  if __name__ == '__main__':     mat = [[5, -2, 2, 7], [1, 0, 0, 3], [-3, 1, 5, 0], [3, -1, -9, 4]]     n = len(mat)     adj = [[0] * n for _ in range(n)]  # To store adjoint      # Print the input matrix     print("Input matrix is:")     for row in mat:         print(row)      # Print the adjoint matrix     print("\nThe Adjoint is:")     adjoint(mat, adj)     for row in adj:         print(row)      # Print the inverse matrix if it exists     print("\nThe Inverse is:")     inv = inverse(mat)     if inv:         for row in inv:             print(row)

using System;  class MatrixOperations {        // Function to get cofactor of mat[p][q] in cof[][]. n is     // current dimension of mat[][]     static void GetCof(int[,] mat, int[,] cof, int p, int q, int n) {         int i = 0, j = 0;         for (int row = 0; row < n; row++) {             for (int col = 0; col < n; col++) {                 if (row != p && col != q) {                     cof[i, j++] = mat[row, col];                     if (j == n - 1) {                         j = 0;                         i++;                     }                 }             }         }     }      // Recursive function for finding determinant of matrix mat of dimension n     static int GetDet(int[,] mat, int n) {         if (n == 1) return mat[0, 0];         int det = 0;          // To store cofactors         int[,] cof = new int[n, n];          int sign = 1;         for (int f = 0; f < n; f++) {             GetCof(mat, cof, 0, f, n);             det += sign * mat[0, f] * GetDet(cof, n - 1);             sign = -sign;         }         return det;     }      // Function to get adjoint of mat in adj     static void Adjoint(int[,] mat, double[,] adj) {         int n = mat.GetLength(0);         if (n == 1) {             adj[0, 0] = 1;             return;         }          int sign = 1;         int[,] cof = new int[n, n];         for (int i = 0; i < n; i++) {             for (int j = 0; j < n; j++) {                 GetCof(mat, cof, i, j, n);                 sign = ((i + j) % 2 == 0) ? 1 : -1;                 adj[j, i] = sign * GetDet(cof, n - 1);             }         }     }      // Function to calculate and store inverse, returns false if matrix is singular     static bool Inverse(int[,] mat, double[,] inv) {         int n = mat.GetLength(0);         int det = GetDet(mat, n);         if (det == 0) {             Console.WriteLine("Singular matrix, can't find its inverse");             return false;         }          double[,] adj = new double[n, n];         Adjoint(mat, adj);          for (int i = 0; i < n; i++)             for (int j = 0; j < n; j++)                 inv[i, j] = adj[i, j] / det;          return true;     }      static void Main() {         int[,] mat = new int[,] { { 5, -2, 2, 7 },                                    { 1, 0, 0, 3 },                                    { -3, 1, 5, 0 },                                    { 3, -1, -9, 4 } };         int n = mat.GetLength(0);         double[,] adj = new double[n, n]; // To store adjoint         double[,] inv = new double[n, n]; // To store inverse          // Print the input matrix         Console.WriteLine("Input matrix is:");         for (int i = 0; i < n; i++) {             for (int j = 0; j < n; j++) {                 Console.Write(mat[i, j] + " ");             }             Console.WriteLine();         }          // Print the adjoint matrix         Console.WriteLine("\nThe Adjoint is:");         Adjoint(mat, adj);         for (int i = 0; i < n; i++) {             for (int j = 0; j < n; j++) {                 Console.Write(adj[i, j] + " ");             }             Console.WriteLine();         }          // Print the inverse matrix if it exists         Console.WriteLine("\nThe Inverse is:");         if (Inverse(mat, inv)) {             for (int i = 0; i < n; i++) {                 for (int j = 0; j < n; j++) {                     Console.Write(inv[i, j] + " ");                 }                 Console.WriteLine();             }         }     } }

JavaScript

// Function to get cofactor of mat[p][q] in cof[][]. n is // current dimension of mat[][] function getCof(mat, cof, p, q, n) {     let i = 0, j = 0;     for (let row = 0; row < n; row++) {         for (let col = 0; col < n; col++) {             if (row !== p && col !== q) {                 cof[i][j++] = mat[row][col];                 if (j === n - 1) {                     j = 0;                     i++;                 }             }         }     } }  // Recursive function for finding determinant  // of matrix mat of dimension n function getDet(mat, n) {     if (n === 1) return mat[0][0];     let det = 0;          // To store cofactors     let cof = Array.from({ length: mat.length }, () => Array(mat.length).fill(0));          let sign = 1;     for (let f = 0; f < n; f++) {         getCof(mat, cof, 0, f, n);         det += sign * mat[0][f] * getDet(cof, n - 1);         sign = -sign;     }     return det; }  // Function to get adjoint of mat in adj function adjoint(mat, adj) {     let n = mat.length;     if (n === 1) {         adj[0][0] = 1;         return;     }          let sign = 1;     let cof = Array.from({ length: n }, () => Array(n).fill(0));     for (let i = 0; i < n; i++) {         for (let j = 0; j < n; j++) {             getCof(mat, cof, i, j, n);             sign = ((i + j) % 2 === 0) ? 1 : -1;             adj[j][i] = sign * getDet(cof, n - 1);         }     } }  // Function to calculate and store inverse, returns  // false if matrix is singular function inverse(mat, inv) {     let n = mat.length;     let det = getDet(mat, n);     if (det === 0) {         console.log("Singular matrix, can't find its inverse");         return false;     }      let adj = Array.from({ length: n }, () => Array(n).fill(0));     adjoint(mat, adj);      for (let i = 0; i < n; i++)         for (let j = 0; j < n; j++)             inv[i][j] = adj[i][j] / det;      return true; }  let mat = [ [ 5, -2, 2, 7 ],             [ 1, 0, 0, 3 ],             [ -3, 1, 5, 0 ],             [ 3, -1, -9, 4 ] ];  let n = mat.length; let adj = Array.from({ length: n }, () => Array(n).fill(0)); // To store adjoint let inv = Array.from({ length: n }, () => Array(n).fill(0)); // To store inverse  // Print the input matrix console.log("Input matrix is:"); mat.forEach(row => {     console.log(row.join(' ')); });  // Print the adjoint matrix console.log("\nThe Adjoint is:"); adjoint(mat, adj); adj.forEach(row => {     console.log(row.join(' ')); });  // Print the inverse matrix if it exists console.log("\nThe Inverse is:"); if (inverse(mat, inv)) {     inv.forEach(row => {         console.log(row.join(' '));     }); }

Output

Input matrix is: 5 -2 2 7  1 0 0 3  -3 1 5 0  3 -1 -9 4   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 determinant of matrix for details of getCofactor() and determinant().

How to inverse a matrix using NumPy

Ashutosh Kumar

Improve

Article Tags :

Practice Tags :

Adjoint and Inverse of a Matrix

Similar Reads

Linear Algebra and Matrix

Statistics for Machine Learning

Probability and Probability Distributions

Calculus for Machine Learning

Regression in Machine Learning