The Determinant of a Matrix is a real number that can be defined for square matrices only i.e, the number of rows and columns of the matrices must be equal. Moreover, it is helpful in determining the system of the linear equation as well as figuring the inverse of the stated matrix.
Procedure to calculate:
- First, we need to calculate the cofactor of all the elements of the matrix in the first row or first column.
- Then, multiply each element of the first row or first column with their respective cofactor.
- At last, we need to add them up with alternate signs.
Example:
- Determinant of 2*2 matrix:
[4, 3] [2, 3] = (4*3)-(3*2) = 12-6 = 6
- Determinant of 3*3 matrix:
[1, 3, -2] [-1, 2, 1] [1, 0, -2] = 1(-4-0)-3(2-1)+(-2)(0-2) = -4-3+4 = -3
Note:
- The determinant of 1*1 matrix is the element itself.
- The Cofactor of any element of the stated matrix can be calculated by eliminating the row and the column of that element from the matrix stated.
Let’s see an example in order to get a clear concept of the above topic.
Example: Using Recursion
Java
// Java program to find// Determinant of a matrixclass 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)); }} |
Output
Determinant of the matrix is : 6
Time complexity: O(n3)
Example: Non-recursion Implementation
Java
// Java program to find Determinant of a matrixclass 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 swapping 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)); }} |
Output
Determinant of the matrix is : 30
Time complexity: O(n3)
