Programa Java para encontrar el determinante de una array

El Determinante de una Array es un número real que se puede definir sólo para arrays cuadradas, es decir, el número de filas y columnas de las arrays debe ser igual. Además, es útil para determinar el sistema de la ecuación lineal, así como para calcular la inversa de la array establecida.

Procedimiento para calcular:

• Primero, necesitamos calcular el cofactor de todos los elementos de la array en la primera fila o primera columna.
• Luego, multiplica cada elemento de la primera fila o primera columna por su respectivo cofactor.
• Por último, necesitamos sumarlos con signos alternativos.

Ejemplo:

• Determinante de la array 2*2:

[4, 3]
[2, 3]

= (4*3)-(3*2)
= 12-6
= 6

• Determinante de la array 3*3:

[1, 3, -2]
[-1, 2, 1]
[1, 0, -2]

= 1(-4-0)-3(2-1)+(-2)(0-2)
= -4-3+4
= -3

Nota:

1. El determinante de la array 1*1 es el propio elemento.
2. El factor C de cualquier elemento de la array indicada se puede calcular eliminando la fila y la columna de ese elemento de la array indicada.

Veamos un ejemplo para tener un concepto claro del tema anterior.

Ejemplo: uso de recursividad

// Java program to find
// Determinant of a matrix
class GFG {
 
    // Dimension of input square matrix
    static final int N = 2;
 
    // Function to get cofactor of
    // mat[p][q] in temp[][]. n is
    // current dimension of mat[][]
    static void getCofactor(int mat[][], 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++] = mat[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 mat[][]. */
    static int determinantOfMatrix(int mat[][], int n)
    {
        int D = 0; // Initialize result
 
        // Base case : if matrix
        // contains single element
        if (n == 1)
            return mat[0][0];
 
        // To store cofactors
        int temp[][] = new int[N][N];
 
        // To store sign multiplier
        int sign = 1;
 
        // Iterate for each element of first row
        for (int f = 0; f < n; f++) {
            // Getting Cofactor of mat[0][f]
            getCofactor(mat, temp, 0, f, n);
            D += sign * mat[0][f]
                 * determinantOfMatrix(temp, n - 1);
 
            // terms are to be added
            // with alternate sign
            sign = -sign;
        }
 
        return D;
    }
 
    /* function for displaying the matrix */
    static void display(int mat[][], int row, int col)
    {
        for (int i = 0; i < row; i++) {
            for (int j = 0; j < col; j++)
                System.out.print(mat[i][j]);
 
            System.out.print("\n");
        }
    }
 
    // Driver code
    public static void main(String[] args)
    {
 
        int mat[][] = { { 4, 3 }, { 2, 3 } };
 
        System.out.print("Determinant "
                         + "of the matrix is : "
                         + determinantOfMatrix(mat, N));
    }
}

Determinant of the matrix is : 6

Complejidad temporal: O(n ³ ) 

Ejemplo: Implementación no recursiva 

// Java program to find Determinant of a matrix
class GFG {
 
    // Dimension of input square matrix
    static final int N = 4;
 
    // Function to get determinant of matrix
    static int determinantOfMatrix(int mat[][], int n)
    {
        int num1, num2, det = 1, index,
                        total = 1; // Initialize result
 
        // temporary array for storing row
        int[] temp = new int[n + 1];
 
        // loop for traversing the diagonal elements
        for (int i = 0; i < n; i++) {
            index = i; // initialize the index
 
            // finding the index which has non zero value
            while (mat[index][i] == 0 && index < n) {
                index++;
            }
            if (index == n) // if there is non zero element
            {
                // the determinant of matrix as zero
                continue;
            }
            if (index != i) {
                // loop for swaping the diagonal element row
                // and index row
                for (int j = 0; j < n; j++) {
                    swap(mat, index, j, i, j);
                }
                // determinant sign changes when we shift
                // rows go through determinant properties
                det = (int)(det * Math.pow(-1, index - i));
            }
 
            // storing the values of diagonal row elements
            for (int j = 0; j < n; j++) {
                temp[j] = mat[i][j];
            }
 
            // traversing every row below the diagonal
            // element
            for (int j = i + 1; j < n; j++) {
                num1 = temp[i]; // value of diagonal element
                num2 = mat[j]
                          [i]; // value of next row element
 
                // traversing every column of row
                // and multiplying to every row
                for (int k = 0; k < n; k++) {
                    // multiplying to make the diagonal
                    // element and next row element equal
                    mat[j][k] = (num1 * mat[j][k])
                                - (num2 * temp[k]);
                }
                total = total * num1; // Det(kA)=kDet(A);
            }
        }
 
        // multiplying the diagonal elements to get
        // determinant
        for (int i = 0; i < n; i++) {
            det = det * mat[i][i];
        }
        return (det / total); // Det(kA)/k=Det(A);
    }
 
    static int[][] swap(int[][] arr, int i1, int j1, int i2,
                        int j2)
    {
        int temp = arr[i1][j1];
        arr[i1][j1] = arr[i2][j2];
        arr[i2][j2] = temp;
        return arr;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int mat[][] = { { 1, 0, 2, -1 },
                        { 3, 0, 0, 5 },
                        { 2, 1, 4, -3 },
                        { 1, 0, 5, 0 } };
 
        // Function call
        System.out.printf(
            "Determinant of the matrix is : %d",
            determinantOfMatrix(mat, N));
    }
}

Determinant of the matrix is : 30

Complejidad temporal: O(n ³ )