The determinant of a Matrix is defined as a special number that is defined only for square matrices (matrices that have the same number of rows and columns). A determinant is used in many places in calculus and other matrices related to algebra, it actually represents the matrix in terms of a real number which can be used in solving a system of a linear equation and finding the inverse of a matrix.
Determinant of 2 x 2 Matrix:
Determinant of 3 x 3 Matrix:
How to calculate?
The value of the determinant of a matrix can be calculated by the following procedure:
- For each element of the first row or first column get the cofactor of those elements.
- Then multiply the element with the determinant of the corresponding cofactor.
- Finally, add them with alternate signs. As a base case, the value of the determinant of a 1*1 matrix is the single value itself.
The cofactor of an element is a matrix that we can get by removing the row and column of that element from that matrix.
Try It Yourself
#include <bits/stdc++.h>
using namespace std;
// Function for finding the determinant of a matrix.
int getDet(vector<vector<int>>& mat, int n) {
// Base case: if the matrix is 1x1
if (n == 1) {
return mat[0][0];
}
// Base case for 2x2 matrix
if (n == 2) {
return mat[0][0] * mat[1][1] -
mat[0][1] * mat[1][0];
}
// Recursive case for larger matrices
int res = 0;
for (int col = 0; col < n; ++col) {
// Create a submatrix by removing the first
// row and the current column
vector<vector<int>> sub(n - 1, vector<int>(n - 1));
for (int i = 1; i < n; ++i) {
int subcol = 0;
for (int j = 0; j < n; ++j) {
// Skip the current column
if (j == col) continue;
// Fill the submatrix
sub[i - 1][subcol++] = mat[i][j];
}
}
// Cofactor expansion
int sign = (col % 2 == 0) ? 1 : -1;
res += sign * mat[0][col] * getDet(sub, n - 1);
}
return res;
}
// Driver program to test the above function
int main() {
vector<vector<int>> mat = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
cout << getDet(mat, mat.size()) << endl;
return 0;
}
#include <stdio.h>
#include <stdlib.h>
#define N 4
// Function for finding the determinant of a matrix.
int getDet(int mat[N][N], int n) {
// Base case: if the matrix is 1x1
if (n == 1) {
return mat[0][0];
}
// Base case for 2x2 matrix
if (n == 2) {
return mat[0][0] * mat[1][1] -
mat[0][1] * mat[1][0];
}
// Recursive case for larger matrices
int res = 0;
for (int col = 0; col < n; ++col) {
// Create a submatrix by removing the
// first row and the current column
int sub[N][N]; // Submatrix
for (int i = 1; i < n; ++i) {
int subcol = 0;
for (int j = 0; j < n; ++j) {
// Skip the current column
if (j == col) continue;
// Fill the submatrix
sub[i - 1][subcol++] = mat[i][j];
}
}
// Cofactor expansion
int sign = (col % 2 == 0) ? 1 : -1;
res += sign * mat[0][col] * getDet(sub, n - 1);
}
return res;
}
// Driver program to test the above function
int main() {
int mat[N][N] = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
printf("%d\n", getDet(mat, N));
return 0;
}
// Function for finding the determinant of a matrix.
public class GfG {
public static int getDet(int[][] mat, int n) {
// Base case: if the matrix is 1x1
if (n == 1) {
return mat[0][0];
}
// Base case for 2x2 matrix
if (n == 2) {
return mat[0][0] * mat[1][1] -
mat[0][1] * mat[1][0];
}
// Recursive case for larger matrices
int res = 0;
for (int col = 0; col < n; ++col) {
// Create a submatrix by removing the first
// row and the current column
int[][] sub = new int[n - 1][n - 1];
for (int i = 1; i < n; ++i) {
int subcol = 0;
for (int j = 0; j < n; ++j) {
// Skip the current column
if (j == col) continue;
// Fill the submatrix
sub[i - 1][subcol++] = mat[i][j];
}
}
// Cofactor expansion
int sign = (col % 2 == 0) ? 1 : -1;
res += sign * mat[0][col] * getDet(sub, n - 1);
}
return res;
}
// Driver program to test the above function
public static void main(String[] args) {
int[][] mat = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
System.out.println(getDet(mat, mat.length));
}
}
# Function for finding the determinant of a matrix.
def getDet(mat, n):
# Base case: if the matrix is 1x1
if n == 1:
return mat[0][0]
# Base case for 2x2 matrix
if n == 2:
return mat[0][0] * mat[1][1] - \
mat[0][1] * mat[1][0]
# Recursive case for larger matrices
res = 0
for col in range(n):
# Create a submatrix by removing the first
# row and the current column
sub = [[0] * (n - 1) for _ in range(n - 1)]
for i in range(1, n):
subcol = 0
for j in range(n):
# Skip the current column
if j == col:
continue
# Fill the submatrix
sub[i - 1][subcol] = mat[i][j]
subcol += 1
# Cofactor expansion
sign = 1 if col % 2 == 0 else -1
res += sign * mat[0][col] * getDet(sub, n - 1)
return res
# Driver program to test the above function
mat = [[1, 0, 2, -1],
[3, 0, 0, 5],
[2, 1, 4, -3],
[1, 0, 5, 0]]
print(getDet(mat, len(mat)))
// Function for finding the determinant of a matrix.
using System;
using System.Linq;
class Determinant {
public static int GetDet(int[,] mat, int n) {
// Base case: if the matrix is 1x1
if (n == 1) {
return mat[0, 0];
}
// Base case for 2x2 matrix
if (n == 2) {
return mat[0, 0] * mat[1, 1] -
mat[0, 1] * mat[1, 0];
}
// Recursive case for larger matrices
int res = 0;
for (int col = 0; col < n; col++) {
// Create a submatrix by removing the first
// row and the current column
int[,] sub = new int[n - 1, n - 1];
for (int i = 1; i < n; i++) {
int subcol = 0;
for (int j = 0; j < n; j++) {
// Skip the current column
if (j == col) continue;
// Fill the submatrix
sub[i - 1, subcol++] = mat[i, j];
}
}
// Cofactor expansion
int sign = (col % 2 == 0) ? 1 : -1;
res += sign * mat[0, col] * GetDet(sub, n - 1);
}
return res;
}
// Driver program to test the above function
static void Main() {
int[,] mat = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
Console.WriteLine(GetDet(mat, mat.GetLength(0)));
}
}
// Function for finding the determinant of a matrix.
function getDet(mat, n) {
// Base case: if the matrix is 1x1
if (n === 1) {
return mat[0][0];
}
// Base case for 2x2 matrix
if (n === 2) {
return mat[0][0] * mat[1][1] -
mat[0][1] * mat[1][0];
}
// Recursive case for larger matrices
let res = 0;
for (let col = 0; col < n; col++) {
// Create a submatrix by removing the first
// row and the current column
let sub = Array.from({ length: n - 1 }, () => new Array(n - 1));
for (let i = 1; i < n; i++) {
let subcol = 0;
for (let j = 0; j < n; j++) {
// Skip the current column
if (j === col) continue;
// Fill the submatrix
sub[i - 1][subcol++] = mat[i][j];
}
}
// Cofactor expansion
let sign = (col % 2 === 0) ? 1 : -1;
res += sign * mat[0][col] * getDet(sub, n - 1);
}
return res;
}
// Driver program to test the above function
let mat = [ [ 1, 0, 2, -1 ],
[ 3, 0, 0, 5 ],
[ 2, 1, 4, -3 ],
[ 1, 0, 5, 0 ] ];
console.log(getDet(mat, mat.length));
Output
30
Time Complexity: O(n3)
Space Complexity: O(n2), Auxiliary space used for storing cofactors.
Note: In the above recursive approach when the size of the matrix is large it consumes more stack size.
Determinant of a Matrix using Determinant properties
We calculates the determinant of an N x N matrix using Gaussian elimination and a series of transformations that reduce the matrix to upper triangular form.
- Converting the given matrix into an upper triangular matrix using determinant properties
- The determinant of the upper triangular matrix is the product of all diagonal elements.
- Iterating every diagonal element and making all the elements down the diagonal as zero using determinant properties
- If the diagonal element is zero then search for the next non-zero element in the same column.
There exist two cases:
- Case 1: If there is no non-zero element. In this case, the determinant of a matrix is zero
- Case 2: If there exists a non-zero element there exist two cases
- Case A: If the index is with a respective diagonal row element. Using the determinant properties make all the column elements down to it zero
- Case B: Swap the row with respect to the diagonal element column and continue the Case A operation.
#include <iostream>
#include <vector>
#include <cmath> // For pow function
using namespace std;
// Function to get determinant of a matrix
int getDet(vector<vector<int>>& mat) {
int n = mat.size();
int num1, num2, det = 1, index, total = 1;
// Temporary array for storing row
vector<int> temp(n + 1);
// Loop for traversing the diagonal elements
for (int i = 0; i < n; i++) {
index = i;
// Finding the index which has non-zero value
while (index < n && mat[index][i] == 0) {
index++;
}
if (index == n) // If there is no non-zero element
{
continue; // The determinant of the matrix is zero
}
if (index != i) {
// Loop for swapping the diagonal element row and index row
for (int j = 0; j < n; j++) {
swap(mat[index][j], mat[i][j]);
}
// Determinant sign changes when we shift rows
det *= 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++) {
// Making the diagonal element and next row element equal
mat[j][k] = (num1 * mat[j][k]) - (num2 * temp[k]);
}
total *= num1;
}
}
// Multiplying the diagonal elements to get determinant
for (int i = 0; i < n; i++) {
det *= mat[i][i];
}
return (det / total); // Det(kA)/k = Det(A);
}
// Driver code
int main() {
vector<vector<int>> mat = {
{ 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 }
};
cout << getDet(mat) << endl;
return 0;
}
import java.util.Arrays;
public class GfG {
// Function to get the determinant of a matrix
static int getDet(int[][] mat) {
int n = mat.length;
int num1, num2, det = 1, index, total = 1;
// 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;
// Finding the index which has a non-zero value
while (index < n && mat[index][i] == 0) {
index++;
}
if (index == n) { // If there is no non-zero element
continue; // The determinant of the matrix is zero
}
if (index != i) {
// Loop for swapping the diagonal element
// row and index row
for (int j = 0; j < n; j++) {
int tempSwap = mat[index][j];
mat[index][j] = mat[i][j];
mat[i][j] = tempSwap;
}
// Determinant sign changes when we shift rows
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++) {
// Making the diagonal element and next row
// element equal
mat[j][k] = (num1 * mat[j][k]) - (num2 * temp[k]);
}
total *= num1;
}
}
// Multiplying the diagonal elements to get determinant
for (int i = 0; i < n; i++) {
det *= mat[i][i];
}
return (det / total); // Det(kA)/k = Det(A);
}
// 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 }
};
System.out.println(getDet(mat));
}
}
# Python program to find Determinant of a matrix
def getDet(mat):
n = len(mat)
temp = [0]*n # temporary array for storing row
total = 1
det = 1 # initialize result
# loop for traversing the diagonal elements
for i in range(0, n):
index = i # initialize the index
# finding the index which has non zero value
while(index < n and mat[index][i] == 0):
index += 1
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 j in range(0, n):
mat[index][j], mat[i][j] = mat[i][j], mat[index][j]
# determinant sign changes when we shift rows
# go through determinant properties
det = det*int(pow(-1, index-i))
# storing the values of diagonal row elements
for j in range(0, n):
temp[j] = mat[i][j]
# traversing every row below the diagonal element
for j in range(i+1, n):
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 k in range(0, n):
# 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 i in range(0, n):
det = det*mat[i][i]
return int(det/total) # Det(kA)/k=Det(A);
# Drivers code
if __name__ == "__main__":
# mat=[[6 1 1][4 -2 5][2 8 7]]
mat = [[1, 0, 2, -1], [3, 0, 0, 5], [2, 1, 4, -3], [1, 0, 5, 0]]
print(getDet(mat))
using System;
class MatrixDeterminant
{
// Function to get the determinant of a matrix
static int getDet(int[,] mat)
{
int n = mat.GetLength(0);
int num1, num2, det = 1, index, total = 1;
// 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;
// Finding the index which has a non-zero value
while (index < n && mat[index, i] == 0)
{
index++;
}
// If there is no non-zero element
if (index == n)
{
// The determinant of the matrix is zero
continue;
}
if (index != i)
{
// Loop for swapping the diagonal element
// row and index row
for (int j = 0; j < n; j++)
{
int tempSwap = mat[index, j];
mat[index, j] = mat[i, j];
mat[i, j] = tempSwap;
}
// Determinant sign changes when we shift rows
det *= (int)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++)
{
// Making the diagonal element and next row
// element equal
mat[j, k] = (num1 * mat[j, k]) - (num2 * temp[k]);
}
total *= num1;
}
}
// Multiplying the diagonal elements to get determinant
for (int i = 0; i < n; i++)
{
det *= mat[i, i];
}
return (det / total); // Det(kA)/k = Det(A);
}
// Driver code
static void Main()
{
int[,] mat = {
{ 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 }
};
Console.WriteLine(getDet(mat));
}
}
// Function to get the determinant of a matrix
function determinantOfMatrix(mat) {
const n = mat.length;
let det = 1;
let total = 1;
// Temporary array for storing row
const temp = new Array(n + 1).fill(0);
// Loop for traversing the diagonal elements
for (let i = 0; i < n; i++) {
let index = i;
// Finding the index which has a non-zero value
while (index < n && mat[index][i] === 0) {
index++;
}
if (index === n) {
continue; // The determinant of the matrix is zero
}
if (index !== i) {
// Swapping the diagonal element row and index row
for (let j = 0; j < n; j++) {
[mat[index][j], mat[i][j]] = [mat[i][j], mat[index][j]];
}
// Determinant sign changes when we shift rows
det *= Math.pow(-1, index - i);
}
// Storing the values of diagonal row elements
for (let j = 0; j < n; j++) {
temp[j] = mat[i][j];
}
// Traversing every row below the diagonal element
for (let j = i + 1; j < n; j++) {
const num1 = temp[i]; // Value of diagonal element
const num2 = mat[j][i]; // Value of next row element
// Traversing every column of row and multiplying
// to every row
for (let k = 0; k < n; k++) {
// Making the diagonal element and next row
// element equal
mat[j][k] = (num1 * mat[j][k]) - (num2 * temp[k]);
}
total *= num1;
}
}
// Multiplying the diagonal elements to get determinant
for (let i = 0; i < n; i++) {
det *= mat[i][i];
}
return (det / total); // Det(kA)/k = Det(A);
}
// Driver code
const mat = [
[1, 0, 2, -1],
[3, 0, 0, 5],
[2, 1, 4, -3],
[1, 0, 5, 0]
];
console.log(determinantOfMatrix(mat));
Output
30
Time complexity: O(n3)
Auxiliary Space: O(n), Space used for storing row.
Determinant of a Matrix using NumPy package in Python
There is a built-in function or method in linalg module of NumPy package in python. It can be called numpy.linalg.det(mat) which returns the determinant value of the matrix mat passed in the argument.