# Java Program to Find the Determinant of a Matrix

• Difficulty Level : Medium
• Last Updated : 08 Sep, 2021

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:

1. The determinant of 1*1 matrix is the element itself.
2. 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 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));``    ``}``}`
Output
`Determinant of the matrix is : 6`

Time complexity: O(n3

Example: Non-recursion Implementation

## Java

 `// 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));``    ``}``}`
Output
`Determinant of the matrix is : 30`

Time complexity: O(n3

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