Given a square matrix, find the adjoint and inverse of the matrix.
We strongly recommend you to refer below as a prerequisite for this.
Determinant of a Matrix
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:
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
- adj(AB) = (adj B).(adj A)
- adj( k A) = kn-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++
// 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 singularbool 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 programint 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;} |
Java
// Java program to find adjoint and inverse of a matrixclass GFG{ static final int N = 4;// Function to get cofactor of A[p][q] in temp[][]. n is current// dimension of A[][]static void getCofactor(int A[][], int temp[][], 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[][]. */static int determinant(int A[][], int n){ int D = 0; // Initialize result // Base case : if matrix contains single element if (n == 1) return A[0][0]; int [][]temp = new int[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].static void adjoint(int A[][],int [][]adj){ if (N == 1) { adj[0][0] = 1; return; } // temp is used to store cofactors of A[][] int sign = 1; int [][]temp = new int[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 singularstatic boolean inverse(int A[][], float [][]inverse){ // Find determinant of A[][] int det = determinant(A, N); if (det == 0) { System.out.print("Singular matrix, can't find its inverse"); return false; } // Find adjoint int [][]adj = new int[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.static void display(int A[][]){ for (int i = 0; i < N; i++) { for (int j = 0; j < N; j++) System.out.print(A[i][j]+ " "); System.out.println(); }}static void display(float A[][]){ for (int i = 0; i < N; i++) { for (int j = 0; j < N; j++) System.out.printf("%.6f ",A[i][j]); System.out.println(); }}// Driver programpublic static void main(String[] args){ int A[][] = { {5, -2, 2, 7}, {1, 0, 0, 3}, {-3, 1, 5, 0}, {3, -1, -9, 4}}; int [][]adj = new int[N][N]; // To store adjoint of A[][] float [][]inv = new float[N][N]; // To store inverse of A[][] System.out.print("Input matrix is :\n"); display(A); System.out.print("\nThe Adjoint is :\n"); adjoint(A, adj); display(adj); System.out.print("\nThe Inverse is :\n"); if (inverse(A, inv)) display(inv);}}// This code is contributed by Rajput-Ji |
Python3
# Python3 program to find adjoint and# inverse of a matrixN = 4# Function to get cofactor of# A[p][q] in temp[][]. n is current# dimension of A[][]def getCofactor(A, temp, p, q, n): i = 0 j = 0 # Looping for each element of the matrix for row in range(n): for col in range(n): # Copying into temporary matrix only those element # which are not in given row and column if (row != p and col != q): temp[i][j] = A[row][col] j += 1 # Row is filled, so increase row index and # reset col index if (j == n - 1): j = 0 i += 1# Recursive function for finding determinant of matrix.# n is current dimension of A[][].def determinant(A, n): D = 0 # Initialize result # Base case : if matrix contains single element if (n == 1): return A[0][0] temp = [] # To store cofactors for i in range(N): temp.append([None for _ in range(N)]) sign = 1 # To store sign multiplier # Iterate for each element of first row for f in range(n): # 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].def adjoint(A, adj): if (N == 1): adj[0][0] = 1 return # temp is used to store cofactors of A[][] sign = 1 temp = [] # To store cofactors for i in range(N): temp.append([None for _ in range(N)]) for i in range(N): for j in range(N): # 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 = [1, -1][(i + j) % 2] # 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 singulardef inverse(A, inverse): # Find determinant of A[][] det = determinant(A, N) if (det == 0): print("Singular matrix, can't find its inverse") return False # Find adjoint adj = [] for i in range(N): adj.append([None for _ in range(N)]) adjoint(A, adj) # Find Inverse using formula "inverse(A) = adj(A)/det(A)" for i in range(N): for j in range(N): inverse[i][j] = adj[i][j] / 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.def display(A): for i in range(N): for j in range(N): print(A[i][j], end=" ") print()def displays(A): for i in range(N): for j in range(N): print(round(A[i][j], 6), end=" ") print()# Driver programA = [[5, -2, 2, 7], [1, 0, 0, 3], [-3, 1, 5, 0], [3, -1, -9, 4]]adj = [None for _ in range(N)]inv = [None for _ in range(N)]for i in range(N): adj[i] = [None for _ in range(N)] inv[i] = [None for _ in range(N)]print("Input matrix is :")display(A)print("\nThe Adjoint is :")adjoint(A, adj)display(adj)print("\nThe Inverse is :")if (inverse(A, inv)): displays(inv)# This code is contributed by phasing17 |
C#
// C# program to find adjoint and inverse of a matrixusing System;using System.Collections.Generic;class GFG{ static readonly int N = 4;// Function to get cofactor of A[p,q] in [,]temp. n is current// dimension of [,]Astatic void getCofactor(int [,]A, int [,]temp, 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. */static int determinant(int [,]A, int n){ int D = 0; // Initialize result // Base case : if matrix contains single element if (n == 1) return A[0, 0]; int [,]temp = new int[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].static void adjoint(int [,]A, int [,]adj){ if (N == 1) { adj[0, 0] = 1; return; } // temp is used to store cofactors of [,]A int sign = 1; int [,]temp = new int[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 singularstatic bool inverse(int [,]A, float [,]inverse){ // Find determinant of [,]A int det = determinant(A, N); if (det == 0) { Console.Write("Singular matrix, can't find its inverse"); return false; } // Find adjoint int [,]adj = new int[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.static void display(int [,]A){ for (int i = 0; i < N; i++) { for (int j = 0; j < N; j++) Console.Write(A[i, j]+ " "); Console.WriteLine(); }}static void display(float [,]A){ for (int i = 0; i < N; i++) { for (int j = 0; j < N; j++) Console.Write("{0:F6} ", A[i, j]); Console.WriteLine(); }}// Driver programpublic static void Main(String[] args){ int [,]A = { {5, -2, 2, 7}, {1, 0, 0, 3}, {-3, 1, 5, 0}, {3, -1, -9, 4}}; int [,]adj = new int[N, N]; // To store adjoint of [,]A float [,]inv = new float[N, N]; // To store inverse of [,]A Console.Write("Input matrix is :\n"); display(A); Console.Write("\nThe Adjoint is :\n"); adjoint(A, adj); display(adj); Console.Write("\nThe Inverse is :\n"); if (inverse(A, inv)) display(inv);}}// This code is contributed by 29AjayKumar |
Javascript
<script>// JavaScript program to find adjoint and // inverse of a matrixlet N = 4;// Function to get cofactor of // A[p][q] in temp[][]. n is current// dimension of A[][]function getCofactor(A,temp,p,q,n){ let i = 0, j = 0; // Looping for each element of the matrix for (let row = 0; row < n; row++) { for (let 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[][]. */function determinant(A,n){ let D = 0; // Initialize result // Base case : if matrix contains single element if (n == 1) return A[0][0]; let temp = new Array(N);// To store cofactors for(let i=0;i<N;i++) { temp[i]=new Array(N); } let sign = 1; // To store sign multiplier // Iterate for each element of first row for (let 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].function adjoint(A,adj){ if (N == 1) { adj[0][0] = 1; return; } // temp is used to store cofactors of A[][] let sign = 1; let temp = new Array(N); for(let i=0;i<N;i++) { temp[i]=new Array(N); } for (let i = 0; i < N; i++) { for (let 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 singularfunction inverse(A,inverse){ // Find determinant of A[][] let det = determinant(A, N); if (det == 0) { document.write("Singular matrix, can't find its inverse"); return false; } // Find adjoint let adj = new Array(N); for(let i=0;i<N;i++) { adj[i]=new Array(N); } adjoint(A, adj); // Find Inverse using formula "inverse(A) = adj(A)/det(A)" for (let i = 0; i < N; i++) for (let j = 0; j < N; j++) inverse[i][j] = adj[i][j]/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.function display(A){ for (let i = 0; i < N; i++) { for (let j = 0; j < N; j++) document.write(A[i][j]+ " "); document.write("<br>"); }}function displays(A){ for (let i = 0; i < N; i++) { for (let j = 0; j < N; j++) document.write(A[i][j].toFixed(6)+" "); document.write("<br>"); }}// Driver programlet A=[[5, -2, 2, 7], [1, 0, 0, 3], [-3, 1, 5, 0], [3, -1, -9, 4]];let adj = new Array(N);let inv = new Array(N);for(let i=0;i<N;i++){ adj[i]=new Array(N); inv[i]=new Array(N);}document.write("Input matrix is :<br>");display(A);document.write("<br>The Adjoint is :<br>");adjoint(A, adj);display(adj);document.write("<br>The Inverse is :<br>");if (inverse(A, inv)) displays(inv);// This code is contributed by rag2127</script> |
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..neveropen.tech/determinant-of-a-matrix/ for details of getCofactor() and determinant().
This article is contributed by Ashutosh Kumar. Please write comments if you find anything incorrect, or if you want to share more information about the topic discussed above.
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