Cramer’s rule: In linear algebra, Cramer’s rule is an explicit formula for the solution of a system of linear equations with as many equations as unknown variables. It expresses the solution in terms of the determinants of the coefficient matrix and of matrices obtained from it by replacing one column by the column vector of the right-hand-sides of the equations. Cramer’s rule is computationally inefficient for systems of more than two or three equations.
Suppose we have to solve these equations:
a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3
Following the Cramer’s Rule, first find the determinant values of all four matrices.
There are 2 cases:
- Case I : When D ≠ 0 In this case we have,
- x = D1/D
- y = D2/D
- z = D3/D
- Hence unique value of x, y, z will be obtained.
- Case II : When D = 0
- When at least one of D1, D2 and D3 is non zero: Then no solution is possible and hence system of equations will be inconsistent.
- When D = 0 and D1 = D2 = D3 = 0: Then the system of equations will be consistent and it will have infinitely many solutions.
Example:
Consider the following system of linear equations.
[2x – y + 3z = 9], [x + y + z = 6], [x – y + z = 2]
[Tex]D_1 = \begin{vmatrix} 9 & -1 & 3\\ 6 & 1 & 1\\ 2 & -1 & 1\\ \end{vmatrix} [/Tex] [Tex]D_3 = \begin{vmatrix} 2 & -1 & 9\\ 1 & 1 & 6\\ 1 & -1 & 2\\ \end{vmatrix} [/Tex] [x = D1/D = 1], [y = D2/D = 2], [z = D3/D = 3]
Below is the implementation.
C++
// CPP program to calculate solutions of linear// equations using cramer's rule#include <bits/stdc++.h>using namespace std;
// This functions finds the determinant of Matrixdouble determinantOfMatrix(double mat[3][3])
{ double ans;
ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])
- mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])
+ mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);
return ans;
}// This function finds the solution of system of// linear equations using cramer's rulevoid findSolution(double coeff[3][4])
{ // Matrix d using coeff as given in cramer's rule
double d[3][3] = {
{ coeff[0][0], coeff[0][1], coeff[0][2] },
{ coeff[1][0], coeff[1][1], coeff[1][2] },
{ coeff[2][0], coeff[2][1], coeff[2][2] },
};
// Matrix d1 using coeff as given in cramer's rule
double d1[3][3] = {
{ coeff[0][3], coeff[0][1], coeff[0][2] },
{ coeff[1][3], coeff[1][1], coeff[1][2] },
{ coeff[2][3], coeff[2][1], coeff[2][2] },
};
// Matrix d2 using coeff as given in cramer's rule
double d2[3][3] = {
{ coeff[0][0], coeff[0][3], coeff[0][2] },
{ coeff[1][0], coeff[1][3], coeff[1][2] },
{ coeff[2][0], coeff[2][3], coeff[2][2] },
};
// Matrix d3 using coeff as given in cramer's rule
double d3[3][3] = {
{ coeff[0][0], coeff[0][1], coeff[0][3] },
{ coeff[1][0], coeff[1][1], coeff[1][3] },
{ coeff[2][0], coeff[2][1], coeff[2][3] },
};
// Calculating Determinant of Matrices d, d1, d2, d3
double D = determinantOfMatrix(d);
double D1 = determinantOfMatrix(d1);
double D2 = determinantOfMatrix(d2);
double D3 = determinantOfMatrix(d3);
printf("D is : %lf \n", D);
printf("D1 is : %lf \n", D1);
printf("D2 is : %lf \n", D2);
printf("D3 is : %lf \n", D3);
// Case 1
if (D != 0) {
// Coeff have a unique solution. Apply Cramer's Rule
double x = D1 / D;
double y = D2 / D;
double z = D3 / D; // calculating z using cramer's rule
printf("Value of x is : %lf\n", x);
printf("Value of y is : %lf\n", y);
printf("Value of z is : %lf\n", z);
}
// Case 2
else {
if (D1 == 0 && D2 == 0 && D3 == 0)
printf("Infinite solutions\n");
else if (D1 != 0 || D2 != 0 || D3 != 0)
printf("No solutions\n");
}
}// Driver Codeint main()
{ // storing coefficients of linear equations in coeff matrix
double coeff[3][4] = {
{ 2, -1, 3, 9 },
{ 1, 1, 1, 6 },
{ 1, -1, 1, 2 },
};
findSolution(coeff);
return 0;
} |
Java
// Java program to calculate solutions of linear// equations using cramer's ruleclass GFG
{// This functions finds the determinant of Matrixstatic double determinantOfMatrix(double mat[][])
{ double ans;
ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])
- mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])
+ mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);
return ans;
}// This function finds the solution of system of// linear equations using cramer's rulestatic void findSolution(double coeff[][])
{ // Matrix d using coeff as given in cramer's rule
double d[][] = {
{ coeff[0][0], coeff[0][1], coeff[0][2] },
{ coeff[1][0], coeff[1][1], coeff[1][2] },
{ coeff[2][0], coeff[2][1], coeff[2][2] },
};
// Matrix d1 using coeff as given in cramer's rule
double d1[][] = {
{ coeff[0][3], coeff[0][1], coeff[0][2] },
{ coeff[1][3], coeff[1][1], coeff[1][2] },
{ coeff[2][3], coeff[2][1], coeff[2][2] },
};
// Matrix d2 using coeff as given in cramer's rule
double d2[][] = {
{ coeff[0][0], coeff[0][3], coeff[0][2] },
{ coeff[1][0], coeff[1][3], coeff[1][2] },
{ coeff[2][0], coeff[2][3], coeff[2][2] },
};
// Matrix d3 using coeff as given in cramer's rule
double d3[][] = {
{ coeff[0][0], coeff[0][1], coeff[0][3] },
{ coeff[1][0], coeff[1][1], coeff[1][3] },
{ coeff[2][0], coeff[2][1], coeff[2][3] },
};
// Calculating Determinant of Matrices d, d1, d2, d3
double D = determinantOfMatrix(d);
double D1 = determinantOfMatrix(d1);
double D2 = determinantOfMatrix(d2);
double D3 = determinantOfMatrix(d3);
System.out.printf("D is : %.6f \n", D);
System.out.printf("D1 is : %.6f \n", D1);
System.out.printf("D2 is : %.6f \n", D2);
System.out.printf("D3 is : %.6f \n", D3);
// Case 1
if (D != 0)
{
// Coeff have a unique solution. Apply Cramer's Rule
double x = D1 / D;
double y = D2 / D;
double z = D3 / D; // calculating z using cramer's rule
System.out.printf("Value of x is : %.6f\n", x);
System.out.printf("Value of y is : %.6f\n", y);
System.out.printf("Value of z is : %.6f\n", z);
}
// Case 2
else
{
if (D1 == 0 && D2 == 0 && D3 == 0)
System.out.printf("Infinite solutions\n");
else if (D1 != 0 || D2 != 0 || D3 != 0)
System.out.printf("No solutions\n");
}
}// Driver Codepublic static void main(String[] args)
{ // storing coefficients of linear
// equations in coeff matrix
double coeff[][] = {{ 2, -1, 3, 9 },
{ 1, 1, 1, 6 },
{ 1, -1, 1, 2 }};
findSolution(coeff);
}
} // This code is contributed by PrinciRaj1992 |
Python3
# Python3 program to calculate # solutions of linear equations # using cramer's rule# This functions finds the # determinant of Matrixdef determinantOfMatrix(mat):
ans = (mat[0][0] * (mat[1][1] * mat[2][2] -
mat[2][1] * mat[1][2]) -
mat[0][1] * (mat[1][0] * mat[2][2] -
mat[1][2] * mat[2][0]) +
mat[0][2] * (mat[1][0] * mat[2][1] -
mat[1][1] * mat[2][0]))
return ans
# This function finds the solution of system of# linear equations using cramer's ruledef findSolution(coeff):
# Matrix d using coeff as given in
# cramer's rule
d = [[coeff[0][0], coeff[0][1], coeff[0][2]],
[coeff[1][0], coeff[1][1], coeff[1][2]],
[coeff[2][0], coeff[2][1], coeff[2][2]]]
# Matrix d1 using coeff as given in
# cramer's rule
d1 = [[coeff[0][3], coeff[0][1], coeff[0][2]],
[coeff[1][3], coeff[1][1], coeff[1][2]],
[coeff[2][3], coeff[2][1], coeff[2][2]]]
# Matrix d2 using coeff as given in
# cramer's rule
d2 = [[coeff[0][0], coeff[0][3], coeff[0][2]],
[coeff[1][0], coeff[1][3], coeff[1][2]],
[coeff[2][0], coeff[2][3], coeff[2][2]]]
# Matrix d3 using coeff as given in
# cramer's rule
d3 = [[coeff[0][0], coeff[0][1], coeff[0][3]],
[coeff[1][0], coeff[1][1], coeff[1][3]],
[coeff[2][0], coeff[2][1], coeff[2][3]]]
# Calculating Determinant of Matrices
# d, d1, d2, d3
D = determinantOfMatrix(d)
D1 = determinantOfMatrix(d1)
D2 = determinantOfMatrix(d2)
D3 = determinantOfMatrix(d3)
print("D is : ", D)
print("D1 is : ", D1)
print("D2 is : ", D2)
print("D3 is : ", D3)
# Case 1
if (D != 0):
# Coeff have a unique solution.
# Apply Cramer's Rule
x = D1 / D
y = D2 / D
# calculating z using cramer's rule
z = D3 / D
print("Value of x is : ", x)
print("Value of y is : ", y)
print("Value of z is : ", z)
# Case 2
else:
if (D1 == 0 and D2 == 0 and
D3 == 0):
print("Infinite solutions")
elif (D1 != 0 or D2 != 0 or
D3 != 0):
print("No solutions")
# Driver Codeif __name__ == "__main__":
# storing coefficients of linear
# equations in coeff matrix
coeff = [[2, -1, 3, 9],
[1, 1, 1, 6],
[1, -1, 1, 2]]
findSolution(coeff)
# This code is contributed by Chitranayal |
C#
// C# program to calculate solutions of linear// equations using cramer's ruleusing System;
class GFG
{// This functions finds the determinant of Matrixstatic double determinantOfMatrix(double [,]mat)
{ double ans;
ans = mat[0,0] * (mat[1,1] * mat[2,2] - mat[2,1] * mat[1,2])
- mat[0,1] * (mat[1,0] * mat[2,2] - mat[1,2] * mat[2,0])
+ mat[0,2] * (mat[1,0] * mat[2,1] - mat[1,1] * mat[2,0]);
return ans;
}// This function finds the solution of system of// linear equations using cramer's rulestatic void findSolution(double [,]coeff)
{ // Matrix d using coeff as given in cramer's rule
double [,]d = {
{ coeff[0,0], coeff[0,1], coeff[0,2] },
{ coeff[1,0], coeff[1,1], coeff[1,2] },
{ coeff[2,0], coeff[2,1], coeff[2,2] },
};
// Matrix d1 using coeff as given in cramer's rule
double [,]d1 = {
{ coeff[0,3], coeff[0,1], coeff[0,2] },
{ coeff[1,3], coeff[1,1], coeff[1,2] },
{ coeff[2,3], coeff[2,1], coeff[2,2] },
};
// Matrix d2 using coeff as given in cramer's rule
double [,]d2 = {
{ coeff[0,0], coeff[0,3], coeff[0,2] },
{ coeff[1,0], coeff[1,3], coeff[1,2] },
{ coeff[2,0], coeff[2,3], coeff[2,2] },
};
// Matrix d3 using coeff as given in cramer's rule
double [,]d3 = {
{ coeff[0,0], coeff[0,1], coeff[0,3] },
{ coeff[1,0], coeff[1,1], coeff[1,3] },
{ coeff[2,0], coeff[2,1], coeff[2,3] },
};
// Calculating Determinant of Matrices d, d1, d2, d3
double D = determinantOfMatrix(d);
double D1 = determinantOfMatrix(d1);
double D2 = determinantOfMatrix(d2);
double D3 = determinantOfMatrix(d3);
Console.Write("D is : {0:F6} \n", D);
Console.Write("D1 is : {0:F6} \n", D1);
Console.Write("D2 is : {0:F6} \n", D2);
Console.Write("D3 is : {0:F6} \n", D3);
// Case 1
if (D != 0)
{
// Coeff have a unique solution. Apply Cramer's Rule
double x = D1 / D;
double y = D2 / D;
double z = D3 / D; // calculating z using cramer's rule
Console.Write("Value of x is : {0:F6}\n", x);
Console.Write("Value of y is : {0:F6}\n", y);
Console.Write("Value of z is : {0:F6}\n", z);
}
// Case 2
else
{
if (D1 == 0 && D2 == 0 && D3 == 0)
Console.Write("Infinite solutions\n");
else if (D1 != 0 || D2 != 0 || D3 != 0)
Console.Write("No solutions\n");
}
}// Driver Codepublic static void Main()
{ // storing coefficients of linear
// equations in coeff matrix
double [,]coeff = {{ 2, -1, 3, 9 },
{ 1, 1, 1, 6 },
{ 1, -1, 1, 2 }};
findSolution(coeff);
}
} // This code is contributed by 29AjayKumar |
Javascript
<script>// Javascript program to calculate solutions of linear// equations using cramer's rule// This functions finds the determinant of Matrixfunction determinantOfMatrix(mat)
{ let ans;
ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])
- mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])
+ mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);
return ans;
}// This function finds the solution of system of// linear equations using cramer's rulefunction findSolution(coeff)
{// Matrix d using coeff as given in cramer's rule let d = [[coeff[0][0], coeff[0][1], coeff[0][2]],
[coeff[1][0], coeff[1][1], coeff[1][2]],
[coeff[2][0], coeff[2][1], coeff[2][2]]];
// Matrix d1 using coeff as given in cramer's rule
let d1 = [[coeff[0][3], coeff[0][1], coeff[0][2]],
[coeff[1][3], coeff[1][1], coeff[1][2]],
[coeff[2][3], coeff[2][1], coeff[2][2]]];
// Matrix d2 using coeff as given in cramer's rule
let d2 = [[coeff[0][0], coeff[0][3], coeff[0][2]],
[coeff[1][0], coeff[1][3], coeff[1][2]],
[coeff[2][0], coeff[2][3], coeff[2][2]]];
// Matrix d3 using coeff as given in cramer's rule
let d3 = [[coeff[0][0], coeff[0][1], coeff[0][3]],
[coeff[1][0], coeff[1][1], coeff[1][3]],
[coeff[2][0], coeff[2][1], coeff[2][3]]];
// Calculating Determinant of Matrices d, d1, d2, d3
let D = determinantOfMatrix(d);
let D1 = determinantOfMatrix(d1);
let D2 = determinantOfMatrix(d2);
let D3 = determinantOfMatrix(d3);
document.write("D is : ", D.toFixed(6)+"<br>");
document.write("D1 is : ", D1.toFixed(6)+"<br>");
document.write("D2 is : ", D2.toFixed(6)+"<br>");
document.write("D3 is : ", D3.toFixed(6)+"<br>");
// Case 1
if (D != 0)
{
// Coeff have a unique solution. Apply Cramer's Rule
let x = D1 / D;
let y = D2 / D;
let z = D3 / D; // calculating z using cramer's rule
document.write("Value of x is : ", x.toFixed(6)+"<br>");
document.write("Value of y is : ", y.toFixed(6)+"<br>");
document.write("Value of z is : ", z.toFixed(6)+"<br>");
}
// Case 2
else
{
if (D1 == 0 && D2 == 0 && D3 == 0)
document.write("Infinite solutions\n");
else if (D1 != 0 || D2 != 0 || D3 != 0)
document.write("No solutions\n");
}
}// Driver Codelet coeff = [[2, -1, 3, 9], [1, 1, 1, 6],
[1, -1, 1, 2]]
findSolution(coeff); // This code is contributed by avanitrachhadiya2155
</script> |
Output
D is : -2.000000 D1 is : -2.000000 D2 is : -4.000000 D3 is : -6.000000 Value of x is : 1.000000 Value of y is : 2.000000 Value of z is : 3.000000
Time complexity: O(1)
Auxiliary space: O(1)