A short note on Cramer’s Rule by three variables

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.

1. Augment AA with the first two columns.
2. 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. 
3. 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: