How to Find the Determinant of a 2x2 Matrix
What is Determinant of a Matrix?
Determinant of a Matrix is a special number that is defined only for square matrices (matrices which have same number of rows and columns). Determinant is used at many places in calculus and other matrix related algebra, it actually represents the matrix in term of a real number which can be used in solving system of linear equation and finding the inverse of a matrix.
How to calculate?
The value of determinant of a matrix can be calculated by following procedure –
For each element of first row or first column get cofactor of those elements and then multiply the element with the determinant of the corresponding cofactor, and finally add them with alternate signs. As a base case the value of determinant of a 1*1 matrix is the single value itself.
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Cofactor of an element, is a matrix which we can get by removing row and column of that element from that matrix.
Determinant of 2 x 2 Matrix:
Determinant of 3 x 3 Matrix:
C++
#include <iostream>
using namespace std;
#define N 4
void getCofactor( int mat[N][N], int temp[N][N], int p,
int q, int n)
{
int i = 0, j = 0;
for ( int row = 0; row < n; row++)
{
for ( int col = 0; col < n; col++)
{
if (row != p && col != q)
{
temp[i][j++] = mat[row][col];
if (j == n - 1)
{
j = 0;
i++;
}
}
}
}
}
int determinantOfMatrix( int mat[N][N], int n)
{
int D = 0;
if (n == 1)
return mat[0][0];
int temp[N][N];
int sign = 1;
for ( int f = 0; f < n; f++)
{
getCofactor(mat, temp, 0, f, n);
D += sign * mat[0][f]
* determinantOfMatrix(temp, n - 1);
sign = -sign;
}
return D;
}
void display( int mat[N][N], int row, int col)
{
for ( int i = 0; i < row; i++)
{
for ( int j = 0; j < col; j++)
cout << " " << mat[i][j];
cout << "n" ;
}
}
int main()
{
int mat[N][N] = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
cout << "Determinant of the matrix is : " << determinantOfMatrix(mat, N);
return 0;
}
C
#include <stdio.h>
#define N 4
void getCofactor( int mat[N][N], int temp[N][N], int p,
int q, int n)
{
int i = 0, j = 0;
for ( int row = 0; row < n; row++)
{
for ( int col = 0; col < n; col++)
{
if (row != p && col != q)
{
temp[i][j++] = mat[row][col];
if (j == n - 1)
{
j = 0;
i++;
}
}
}
}
}
int determinantOfMatrix( int mat[N][N], int n)
{
int D = 0;
if (n == 1)
return mat[0][0];
int temp[N][N];
int sign = 1;
for ( int f = 0; f < n; f++)
{
getCofactor(mat, temp, 0, f, n);
D += sign * mat[0][f]
* determinantOfMatrix(temp, n - 1);
sign = -sign;
}
return D;
}
void display( int mat[N][N], int row, int col)
{
for ( int i = 0; i < row; i++)
{
for ( int j = 0; j < col; j++)
printf ( " %d" , mat[i][j]);
printf ( "n" );
}
}
int main()
{
int mat[N][N] = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
printf ( "Determinant of the matrix is : %d" ,
determinantOfMatrix(mat, N));
return 0;
}
Java
class GFG {
static final int N = 4 ;
static void getCofactor( int mat[][], int temp[][],
int p, int q, int n)
{
int i = 0 , j = 0 ;
for ( int row = 0 ; row < n; row++)
{
for ( int col = 0 ; col < n; col++)
{
if (row != p && col != q)
{
temp[i][j++] = mat[row][col];
if (j == n - 1 )
{
j = 0 ;
i++;
}
}
}
}
}
static int determinantOfMatrix( int mat[][], int n)
{
int D = 0 ;
if (n == 1 )
return mat[ 0 ][ 0 ];
int temp[][] = new int [N][N];
int sign = 1 ;
for ( int f = 0 ; f < n; f++)
{
getCofactor(mat, temp, 0 , f, n);
D += sign * mat[ 0 ][f]
* determinantOfMatrix(temp, n - 1 );
sign = -sign;
}
return D;
}
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" );
}
}
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.print( "Determinant "
+ "of the matrix is : "
+ determinantOfMatrix(mat, N));
}
}
Python3
def getcofactor(m, i, j):
return [row[: j] + row[j + 1 :] for row in (m[: i] + m[i + 1 :])]
def determinantOfMatrix(mat):
if ( len (mat) = = 2 ):
value = mat[ 0 ][ 0 ] * mat[ 1 ][ 1 ] - mat[ 1 ][ 0 ] * mat[ 0 ][ 1 ]
return value
Sum = 0
for current_column in range ( len (mat)):
sign = ( - 1 ) * * (current_column)
sub_det = determinantOfMatrix(getcofactor(mat, 0 , current_column))
Sum + = (sign * mat[ 0 ][current_column] * sub_det)
return Sum
if __name__ = = '__main__' :
mat = [[ 1 , 0 , 2 , - 1 ],
[ 3 , 0 , 0 , 5 ],
[ 2 , 1 , 4 , - 3 ],
[ 1 , 0 , 5 , 0 ]]
print ( 'Determinant of the matrix is :' , determinantOfMatrix(mat))
C#
using System;
class GFG {
static int N = 4;
static void getCofactor( int [, ] mat, int [, ] temp,
int p, int q, int n)
{
int i = 0, j = 0;
for ( int row = 0; row < n; row++)
{
for ( int col = 0; col < n; col++)
{
if (row != p && col != q) {
temp[i, j++] = mat[row, col];
if (j == n - 1) {
j = 0;
i++;
}
}
}
}
}
static int determinantOfMatrix( int [, ] mat, int n)
{
int D = 0;
if (n == 1)
return mat[0, 0];
int [, ] temp = new int [N, N];
int sign = 1;
for ( int f = 0; f < n; f++)
{
getCofactor(mat, temp, 0, f, n);
D += sign * mat[0, f]
* determinantOfMatrix(temp, n - 1);
sign = -sign;
}
return D;
}
static void display( int [, ] mat, int row, int col)
{
for ( int i = 0; i < row; i++)
{
for ( int j = 0; j < col; j++)
Console.Write(mat[i, j]);
Console.Write( "\n" );
}
}
public static void Main()
{
int [, ] mat = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
Console.Write( "Determinant "
+ "of the matrix is : "
+ determinantOfMatrix(mat, N));
}
}
Javascript
<script>
let N = 4;
function getCofactor(mat,temp,p,q,n)
{
let i = 0, j = 0;
for (let row = 0; row < n; row++)
{
for (let col = 0; col < n; col++)
{
if (row != p && col != q)
{
temp[i][j++] = mat[row][col];
if (j == n - 1)
{
j = 0;
i++;
}
}
}
}
}
function determinantOfMatrix(mat,n)
{
let D = 0;
if (n == 1)
return mat[0][0];
let temp = new Array(N);
for (let i=0;i<N;i++)
{
temp[i]= new Array(N);
}
let sign = 1;
for (let f = 0; f < n; f++)
{
getCofactor(mat, temp, 0, f, n);
D += sign * mat[0][f]
* determinantOfMatrix(temp, n - 1);
sign = -sign;
}
return D;
}
function display(mat,row,col)
{
for (let i = 0; i < row; i++)
{
for (let j = 0; j < col; j++)
document.write(mat[i][j]);
document.write( "<br>" );
}
}
let mat=[[ 1, 0, 2, -1 ],
[ 3, 0, 0, 5 ],
[ 2, 1, 4, -3 ],
[ 1, 0, 5, 0 ]];
document.write( "Determinant "
+ "of the matrix is : "
+ determinantOfMatrix(mat, N));
</script>
Output
Determinant of the matrix is : 30
Adjoint and Inverse of a Matrix
There are various properties of the Determinant which can be helpful for solving problems related with matrices,
This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
In Above Method Recursive Approach is discussed. When the size of matrix is large it consumes more stack size
In this Method We are using the properties of Determinant. In this approach we are converting the given matrix into upper triangular matrix using determinant properties The determinant of upper triangular matrix is the product of all diagonal elements For properties on determinant go through this website https://cran.r-project.org/web/packages/matlib/vignettes/det-ex1.html
In this approach, we are iterating every diagonal element and making all the elements down the diagonal as zero using determinant properties
If the diagonal element is zero then we will search 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 matrix is zero
Case 2:
If there exists non-zero element there exist two cases
Case a:
if index is with respective diagonal row element. Using the determinant properties we make all the column elements down to it as zero
Case b:
Here we need to swap the row with respective to diagonal element column and continue the case 'a; operation
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>
using namespace std;
#define N 4
int determinantOfMatrix( int mat[N][N], int n)
{
int num1, num2, det = 1, index,
total = 1;
int temp[n + 1];
for ( int i = 0; i < n; i++)
{
index = i;
while (mat[index][i] == 0 && index < n)
{
index++;
}
if (index == n)
{
continue ;
}
if (index != i)
{
for ( int j = 0; j < n; j++)
{
swap(mat[index][j], mat[i][j]);
}
det = det * pow (-1, index - i);
}
for ( int j = 0; j < n; j++)
{
temp[j] = mat[i][j];
}
for ( int j = i + 1; j < n; j++)
{
num1 = temp[i];
num2 = mat[j][i];
for ( int k = 0; k < n; k++)
{
mat[j][k]
= (num1 * mat[j][k]) - (num2 * temp[k]);
}
total = total * num1;
}
}
for ( int i = 0; i < n; i++)
{
det = det * mat[i][i];
}
return (det / total);
}
int main()
{
int mat[N][N] = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
printf ( "Determinant of the matrix is : %d" ,
determinantOfMatrix(mat, N));
return 0;
}
Java
class GFG
{
static final int N = 4 ;
static int determinantOfMatrix( int mat[][], int n)
{
int num1, num2, det = 1 , index,
total = 1 ;
int [] temp = new int [n + 1 ];
for ( int i = 0 ; i < n; i++)
{
index = i;
while (mat[index][i] == 0 && index < n)
{
index++;
}
if (index == n)
{
continue ;
}
if (index != i)
{
for ( int j = 0 ; j < n; j++)
{
swap(mat, index, j, i, j);
}
det = ( int )(det * Math.pow(- 1 , index - i));
}
for ( int j = 0 ; j < n; j++)
{
temp[j] = mat[i][j];
}
for ( int j = i + 1 ; j < n; j++)
{
num1 = temp[i];
num2 = mat[j]
[i];
for ( int k = 0 ; k < n; k++)
{
mat[j][k] = (num1 * mat[j][k])
- (num2 * temp[k]);
}
total = total * num1;
}
}
for ( int i = 0 ; i < n; i++)
{
det = det * mat[i][i];
}
return (det / total);
}
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;
}
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.printf(
"Determinant of the matrix is : %d" ,
determinantOfMatrix(mat, N));
}
}
Python3
def determinantOfMatrix(mat, n):
temp = [ 0 ] * n
total = 1
det = 1
for i in range ( 0 , n):
index = i
while (mat[index][i] = = 0 and index < n):
index + = 1
if (index = = n):
continue
if (index ! = i):
for j in range ( 0 , n):
mat[index][j], mat[i][j] = mat[i][j], mat[index][j]
det = det * int ( pow ( - 1 , index - i))
for j in range ( 0 , n):
temp[j] = mat[i][j]
for j in range (i + 1 , n):
num1 = temp[i]
num2 = mat[j][i]
for k in range ( 0 , n):
mat[j][k] = (num1 * mat[j][k]) - (num2 * temp[k])
total = total * num1
for i in range ( 0 , n):
det = det * mat[i][i]
return int (det / total)
if __name__ = = "__main__" :
mat = [[ 1 , 0 , 2 , - 1 ], [ 3 , 0 , 0 , 5 ], [ 2 , 1 , 4 , - 3 ], [ 1 , 0 , 5 , 0 ]]
N = len (mat)
print ( "Determinant of the matrix is : " , determinantOfMatrix(mat, N))
C#
using System;
class GFG {
static readonly int N = 4;
static int determinantOfMatrix( int [, ] mat, int n)
{
int num1, num2, det = 1, index,
total = 1;
int [] temp = new int [n + 1];
for ( int i = 0; i < n; i++)
{
index = i;
while (mat[index, i] == 0 && index < n)
{
index++;
}
if (index == n)
{
continue ;
}
if (index != i)
{
for ( int j = 0; j < n; j++)
{
swap(mat, index, j, i, j);
}
det = ( int )(det * Math.Pow(-1, index - i));
}
for ( int j = 0; j < n; j++)
{
temp[j] = mat[i, j];
}
for ( int j = i + 1; j < n; j++)
{
num1 = temp[i];
num2 = mat[j,
i];
for ( int k = 0; k < n; k++)
{
mat[j, k] = (num1 * mat[j, k])
- (num2 * temp[k]);
}
total = total * num1;
}
}
for ( int i = 0; i < n; i++)
{
det = det * mat[i, i];
}
return (det / total);
}
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;
}
public static void Main(String[] args)
{
int [, ] mat = { { 1, 0, 2, -1 },
{ 3, 0, 0, 5 },
{ 2, 1, 4, -3 },
{ 1, 0, 5, 0 } };
Console.Write( "Determinant of the matrix is : {0}" ,
determinantOfMatrix(mat, N));
}
}
Javascript
Javascript<script>
var N = 4;
function determinantOfMatrix(mat , n)
{
var num1, num2, det = 1, index,
total = 1;
var temp = Array(n + 1).fill(0);
for (i = 0; i < n; i++)
{
index = i;
while (mat[index][i] == 0 && index < n)
{
index++;
}
if (index == n)
{
continue ;
}
if (index != i)
{
for (j = 0; j < n; j++)
{
swap(mat, index, j, i, j);
}
det = parseInt((det * Math.pow(-1, index - i)));
}
for (j = 0; j < n; j++)
{
temp[j] = mat[i][j];
}
for (j = i + 1; j < n; j++)
{
num1 = temp[i];
num2 = mat[j]
[i];
for (k = 0; k < n; k++)
{
mat[j][k] = (num1 * mat[j][k])
- (num2 * temp[k]);
}
total = total * num1;
}
}
for (i = 0; i < n; i++)
{
det = det * mat[i][i];
}
return (det / total);
}
function swap(arr , i1 , j1 , i2,
j2)
{
var temp = arr[i1][j1];
arr[i1][j1] = arr[i2][j2];
arr[i2][j2] = temp;
return arr;
}
var mat = [ [ 1, 0, 2, -1 ],
[ 3, 0, 0, 5 ],
[ 2, 1, 4, -3 ],
[ 1, 0, 5, 0 ] ];
document.write(
"Determinant of the matrix is : " ,
determinantOfMatrix(mat, N));
</script>
Output
Determinant of the matrix is : 30
Time complexity: O(n3)
Auxiliary Space: O(n)
Method 3: 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 as numpy.linalg.det(mat) which returns the determinant value of matrix mat passed in the argument.
Python3
import numpy as np
def determinant(mat):
det = np.linalg.det(mat)
return round (det)
mat = [[ 1 , 0 , 2 , - 1 ],
[ 3 , 0 , 0 , 5 ],
[ 2 , 1 , 4 , - 3 ],
[ 1 , 0 , 5 , 0 ]]
print ( 'Determinant of the matrix is:' ,
determinant(mat))
Output:
Determinant of the matrix is: 30.0
How to Find the Determinant of a 2x2 Matrix
Source: https://www.geeksforgeeks.org/determinant-of-a-matrix/
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