Dada una array cuadrada, encuentre la adjunta y la inversa de la array.
Le recomendamos encarecidamente que consulte a continuación como un requisito previo de esto.
Determinante de una array
¿Qué es adjunto?
La array adjunta (o adjunta) de una array es la array obtenida al tomar la transposición de la array cofactor de una array cuadrada dada y se denomina array adjunta o adjunta. El adjunto de cualquier array cuadrada ‘A’ (por ejemplo) se representa como Adj(A).
Ejemplo:
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]
Propiedades importantes:
- El producto de una array cuadrada A con su adjunta produce una array diagonal, donde cada entrada diagonal es igual al determinante de A.
es decir,
A.adj(A) = det(A).I I => Identity matrix of same order as of A. det(A) => Determinant value of A
- Se dice que una array cuadrada distinta de cero ‘A’ de orden n es invertible si existe una única array cuadrada ‘B’ de orden n tal que,
A.B = B.A = I The matrix 'B' is said to be inverse of 'A'. i.e., B = A-1
¿Cómo encontrar adjunto?
Seguimos la definición dada arriba.
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]
¿Cómo encontrar la inversa?
El inverso de una array existe solo si la array no es singular, es decir, el determinante no debe ser 0.
Usando determinante y adjunto, podemos encontrar fácilmente el inverso de una array cuadrada usando la siguiente fórmula,
If det(A) != 0
A-1 = adj(A)/det(A)
Else
"Inverse doesn't exist"
La inversa se utiliza para encontrar la solución a un sistema de ecuación lineal.
A continuación se muestra la implementación para encontrar adjuntos e inversos de una array.
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 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;
}
Java
// Java program to find adjoint and inverse of a matrix
class 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 singular
static 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 program
public 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 matrix
N = 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 singular
def 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 program
A = [[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 matrix
using 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 [,]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 singular
static 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 program
public 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 matrix
let 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 singular
function 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 program
let 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>
Producción:
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
Consulte https://www.geeksforgeeks.org/determinant-of-a-matrix/ para obtener detalles de getCofactor() y determinante().
Referencias:
https://www.geeksforgeeks.org/determinant-of-a-matrix/
https://en.wikipedia.org/wiki/Adjugate_matrix Ashutosh Kumar
contribuye con este artículo . Escriba comentarios si encuentra algo incorrecto o si desea compartir más información sobre el tema tratado anteriormente.
Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA