Adjoint and Inverse of a Matrix (original) (raw)
Last Updated : 27 Feb, 2026
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:
**Input: A[][]= [[5, -2, 2, 7], [1, 0, 0, 3], [-3, 1, 5, 0], [3, -1, -9, 4]]
**Output:
Adj(A) = [[-12, 76, -60, -36], [-56, 208, -82, -58], [4, 4, -2, -10], [4, 4, 20, 12]],
Inverse(A) = [[-0.136, 0.864, -0.682, -0.409], [-0.636, 2.364, -0.932, -0.659], [0.045, 0.045, -0.023, -0.114], [0.045, 0.045, 0.227, 0.136]]
**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. If 'I' represent the Identity matrix of same order as of A and det(A) represent the determinant value of A. Then,
A.adj(A) = det(A).I
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
Here ,
- 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 of A ?
We follow the definition given above.
Let A[N][N] be input matrix.
- Create a matrix adj[N][N] store the adjoint matrix.
- 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"
**Note: Inverse is used to find the solution to a system of linear equations.
C++ `
#include #include using namespace std;
// Function to get cofactor of mat[p][q] in cof[][]. n is // current dimension of mat[][] void getCof(vector<vector>& mat, vector<vector>& 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>& 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>& mat, vector<vector>& 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>& mat, vector<vector>& 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> 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));
vector<vector<float>> inv(n, vector<float>(n));
cout << "\nThe Adjoint is:\n";
adjoint(mat, adj);
for (auto& row : adj) {
for (int val : row) cout << val << " ";
cout << endl;
}
cout << "\nThe Inverse is:\n";
if (inverse(mat, inv)) {
for (auto& row : inv) {
for (float val : row){
cout << fixed << setprecision(3)<<val << " ";
}
cout << endl;
}
}
return 0;}
Java
public class GfG {
// 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;
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[][] 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);
int 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");
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] = adj[i][j] / (float) 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;
// Adjoint matrix
int[][] adj = new int[n][n];
adjoint(mat, adj);
System.out.println("The Adjoint is:");
for (int[] row : adj) {
for (int val : row)
System.out.printf("%.3f ", (float)val);
System.out.println();
}
// Inverse matrix
float[][] inv = new float[n][n];
System.out.println("\nThe Inverse is:");
if (inverse(mat, inv)) {
for (float[] row : inv) {
for (float val : row)
System.out.printf("%.3f ", val);
System.out.println();
}
}
}}
Python
Function to get cofactor of mat[p][q] in cof[][]. n is
current dimension of mat[][]
def get_cof(mat, p, q, n): cof = [] for i in range(n): if i == p: continue row = [] for j in range(n): if j == q: continue row.append(mat[i][j]) cof.append(row) return cof
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
sign = 1
for f in range(n):
cof = get_cof(mat, 0, f, n)
det += sign * mat[0][f] * get_det(cof, n-1)
sign = -sign
return detFunction to get adjoint of mat in adj
def adjoint(mat): n = len(mat) adj = [[0]*n for _ in range(n)] for i in range(n): for j in range(n): cof = get_cof(mat, i, j, n) sign = 1 if (i+j)%2==0 else -1 adj[j][i] = sign * get_det(cof, n-1) return adj
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: return None
adj = adjoint(mat)
inv = [[adj[i][j]/det for j in range(n)] for i in range(n)]
return adj, invif name == "main": mat = [ [5, -2, 2, 7], [1, 0, 0, 3], [-3, 1, 5, 0], [3, -1, -9, 4] ]
result = inverse(mat)
if result:
adj, inv = result
print("Adjoint:")
for row in adj:
print(" ".join(f"{float(val):.3f}" for val in row))
print("\nInverse:")
for row in inv:
print(" ".join(f"{val:.3f}" for val in row))
else:
print("Singular matrix")C#
using System;
class GfG {
// 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, sign = 1;
int[,] cof = new int[n,n];
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) {
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);
int 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, float[,] inv, int n) {
int det = GetDet(mat, n);
if (det == 0) return false;
int[,] adj = new int[n,n];
Adjoint(mat, adj, n);
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
inv[i,j] = adj[i,j] / (float)det;
return true;
}
static void Main() {
int[,] mat = {
{5, -2, 2, 7},
{1, 0, 0, 3},
{-3, 1, 5, 0},
{3, -1, -9, 4}
};
int n = 4;
int[,] adj = new int[n,n];
Adjoint(mat, adj, n);
Console.WriteLine("Adjoint:");
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++)
Console.Write("{0:F3} ", (float)adj[i,j]);
Console.WriteLine();
}
float[,] inv = new float[n,n];
Console.WriteLine("\nInverse:");
if (Inverse(mat, inv, n)) {
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++)
Console.Write("{0:F3} ", inv[i,j]);
Console.WriteLine();
}
}
else Console.WriteLine("Singular matrix");
}}
JavaScript
// Function to get cofactor of mat[p][q] in cof[][]. n is // current dimension of mat[][] function getCof(mat, p, q, n) { let cof = []; for (let i = 0; i < n; i++) { if (i === p) continue; let row = []; for (let j = 0; j < n; j++) { if (j === q) continue; row.push(mat[i][j]); } cof.push(row); } return cof; }
// 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, sign = 1;
for (let f = 0; f < n; f++) {
let cof = getCof(mat, 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) { let n = mat.length; let adj = Array.from({length:n}, () => Array(n).fill(0));
for (let i = 0; i < n; i++)
for (let j = 0; j < n; j++) {
let cof = getCof(mat, i, j, n);
let sign = ((i+j)%2===0)?1:-1;
adj[j][i] = sign * getDet(cof, n-1);
}
return adj;}
// Function to calculate and store inverse, returns // false if matrix is singular function inverse(mat) { let n = mat.length; let det = getDet(mat, n); if (det === 0) return null;
let adj = adjoint(mat);
let inv = Array.from({length:n}, () => Array(n));
for (let i = 0; i < n; i++)
for (let j = 0; j < n; j++)
inv[i][j] = adj[i][j] / det;
return {adj, inv};}
//Driver code let mat = [ [5, -2, 2, 7], [1, 0, 0, 3], [-3, 1, 5, 0], [3, -1, -9, 4] ];
let result = inverse(mat);
if (result) { console.log("Adjoint:"); result.adj.forEach(row => console.log(row.map(v => v.toFixed(3)).join(" ")) );
console.log("\nInverse:");
result.inv.forEach(row =>
console.log(row.map(v => v.toFixed(3)).join(" "))
);} else { console.log("Singular matrix"); }
`
Output
The Adjoint is: -12 76 -60 -36 -56 208 -82 -58 4 4 -2 -10 4 4 20 12
The Inverse is: -0.136 0.864 -0.682 -0.409 -0.636 2.364 -0.932 -0.659 0.045 0.045 -0.023 -0.114 0.045 0.045 0.227 0.136
**Note: Refer to the determinant of a matrix section for detailed information about the getCofactor() and determinant() functions.