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Introduction
This Rule is majorly used to find the solution of equations with any number of variables. Sometimes what happens when we solve a system of equations in 3 variables, say x, y, and z. Here, we may need to solve for two variables, x, and y, to solve our third variable, z. But using this new Cramer’s Rule, we can find the value of any variable without finding the other variables’ values and calculate the inverse of a matrix, or you can say calculator inverse matrix. We don’t have to depend on other variables to get the value of a particular third variable. But this Rule has some limitations concerning the solutions.
Cramer’s Rule by three variables
This name is called the Cramer’s Rule because of its publisher’s name, Gabriel Cramer (1704–1752), who is very famous as he published this Rule in 1750. This formula is the most commonly used to get the solution for the given system of equations in the form of matrices. The solution is calculated using Cramer’s Rule, which will be in terms of the determinants of the matrix only. Cramer’s Rule is one of the important and easiest methods applied to solve a system of equations most simply. In this method, the variable’s value is calculated using the determinants of matrices.
Thus, the second name of the Cramer’s Rule is the determinant method. Think of a linear equation written in the matrix form AX = B.
Here,
A = Coefficient matrix
X = Column matrix with variables
B = Column matrix with the constants
Now, we have to find the determinants as
D = |A|, Dx1, Dx2, Dx3,…, Dxn
Here,
Dxi for i = 1, 2, 3,…, n is the same determinant as D such that the column is replaced with B.
Thus,
x1 = Dx1/D; x2 = Dx2/D; x3 = Dx3/D; ….; xn = Dxn/D {where D is not equal to 0}Let’s have a look at the formulas of Cramer’s Rule for 3×3 matrices.
Find the determinant of the 3×3 matrix.
- Augment AA with the first two columns.
- Upper left to lower right: simply, multiply the entries down the first diagonal. Now add the result to the product of entries down the second diagonal.
- From lower left to upper right: First of all, subtract the product of entries up the first diagonal. And from this result, subtract the product of entries up the second diagonal. You can now minus or subtract the product of entries up the third diagonal from this result.
The algebra is as follows:
Now have a look at the following example. We are trying to explain to you a problem.
Now we will tell you how to calculate the inverse of a matrix.
To calculate Inverse of Matrix for a matrix, A is written by A-1. The calculated inverse matrix of a two × two matrix can be calculated using a simple formula. Further, to find the inverse of a three × three matrix, we need to know about the determinant and adjoint of the matrix. The calculated inverse matrix is another kind of matrix. When we multiply that with the given matrix, it gives the multiplicative identity.
To calculate the inverse of the matrix: This Cramer’s rule method is used to calculate a normal or the linear kind of equation in the form of the matrix in inversion form. Now, let us learn about the formula and its methods related to the calculated inverse of the matrix. To find the inverse of a 3×3 matrix, first of all, calculate the determinant of the matrix. Now check, if the determinant is 0, by chance, then the matrix has no inverse. This is the Rule, and now you can change the matrix by rewriting the first row as the first column, the middle row as the middle column, and the third row as the third column. Change all rows in all columns.
Conclusion
Cramer’s Rule is a very efficient and perfect method to find the solutions in the matrix. Here, it is provided that we have the same number of equations as unknowns. This Cramer’s Rule will give us the unique solution to a system of all the equations if it exists. Unlike normal equations here, we don’t have to be dependent on other variables to know the value of the third variable. Cramer’s rule is a method to solve the equations but in the form of a matrix, where there are the same amount of unknowns as equations in the system.
