Cramer’s Rule

In algebra, Cramer’s rule is defined as an explicit formula or method used to solve a system or series of linear equations. It applies to those linear equations, having as many unknown variables as values and when a unique solution is probable. The Cramer rule method uses determinants to find the values of the variables given in a linear equation. The rule expresses the solution in terms of determinants by arranging the matrix in the form of Ax=B, where:

• A represents the coefficient matrix and contains all the numerical values.

• X represents the matrix of variables.

• B represents the matrix with all the constants on the right-hand side of the equation.

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History of Cramer’s Rule

Cramer's rule was formed and developed by the great mathematician Gabriel Cramer in the 1750s. He used Cramer’s method to find the solution of a system of equations with n number of variables and the same number of equations. 

Cramer’s Rule Formula

Cramer’s rule formula is easy to understand, and it helps us solve the AX=B form of the matrices. With the help of the formula, we can find the values of n number of variables in a linear equation. To solve using Cramer’s rule the equation AX=B, we need to follow the steps below:

• To solve by Cramer's rule, we need to find the determinants for matrix A, x1, x2, x3, ...., xn. 

• Each of these determinant matrices is denoted by D, such as D = |A|, Dx1, Dx2, Dx3, …, Dxn. 

• Here the D represents the determinant values of |A|, and Dxn represents the determinant values of the x matrix, where values of B replace the nth column. 

• Once we get all the determinant values for the equations, we apply the formula:

• Xn=  Dxn/D. We can apply this formula to as many equations as present in the matrices. Always remember that the value of D should never be equal to zero (D≠0).

• Hence, with the formula mentioned above, we can derive the value of the variables using Cramer’s method.

With Cramer’s rule formula, we can solve the system of two equations in two variables and three equations in three variables. If you are wondering how below, we will discuss the method in detail. 

Cramer’s Rule 2x2

Using Cramer’s rule formula, we will solve two equations in two variables. Let’s discuss the steps to solve two equations in two variables using Cramer's rule matrix. For Cramer’s rule 2x2, let’s consider two equations in 2 variables, x and y:

Let the equation be:

p1x + q1y = r1 and,

p2x + q2y = r2.

Now, follow the steps below to solve using Cramer’s rule:

• Step 1 - Write the complete equation using Cramer’s rule matrix in the form of Ax=B, where: 

A =   $\begin{bmatrix}p₁ &q₁ \\ p₂ &q₂ \end{bmatrix}$

x =   $\begin{bmatrix}X\\ Y\end{bmatrix}$

And,

B =    $\begin{bmatrix}r₁\\ r₂ \end{bmatrix}$

• Step 2 - Here, we need to find the determinants of the equations, where D = |A|, and Dx and Dy.

For D or |A|, solve :

$\begin{vmatrix}p₁ &q₁\\ p₂ &q₂  \end{vmatrix}$

For DX, replace the first column of matrix A with values of B; hence we get Dx:

$\begin{vmatrix}r₁ &q₁\\ r₂ &q₂  \end{vmatrix}$

For Dy, replace the second column of matrix A with values of B and solve

$\begin{vmatrix}p₁ &r₁\\ p₂ &r₂  \end{vmatrix}$

• Step 3 - Once we get the determinants of A, Dx, and Dy, we will find values of x and y using:

X = DX/D

Y = Dy/D.

Cramer’s Rule 3x3

Above, we discussed the methods and steps to solve 2x2 equations using Cramer’s rule. Now, we will discuss solving three equations in three variables by Cramer’s method. For solving Cramer’s rule, 3x3 equations consider an equation with three variables x, y, and z.

Let the equation be:

p1 x + q1y + r1z = s1

P2x + q2y + r2z = s2

p3x + q3y + r3z = s3

Now, follow the steps mentioned below to solve by Cramer’s rule:

• Step 1 - First, write the equation in the form AX=B, where

A =   $\begin{bmatrix}p₁ &q₁  &r₁ \\ p₂ &q₂  &r₂ \\ p₃ &q₃  &r₃ \end{bmatrix}$

X =    $\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}$

B =    $\begin{bmatrix}s₁ \\ s₂\\ s₃ \end{bmatrix}$

• As per Cramer’s rule, determinant value is important to find the value of variables. Here also we need to find D|A|, Dx, Dy, and Dz. Where:

D|A| =  $\begin{vmatrix}p₁  &q₁   &r₁  \\ p₂ &q₂  &r₂ \\ p₃ &q₃  &r₃\end{vmatrix}$

For Dx, replace the first column of matrix A with values of B.

Dx =  $\begin{vmatrix}s₁  &q₁   &r₁  \\ s₂&q₂ &r₂\\ s₃&q₃ &r₃\end{vmatrix}$

For Dy, replace the second column of matrix A with values of B.

Dy, = $\begin{vmatrix}p₁ &s₁  &r₁ \\ p₂ &s₂ &r₂ \\ p₃ &s₃  &r₃\end{vmatrix}$

For Dz, replace the third column of matrix A with the values of B.

Dz =  $\begin{vmatrix}p₁ &q₁  &s₁ \\ p₂ &q₂  &s₂ \\ p₃ &q₃  &s₃ \end{vmatrix}$

• Step 2 - Once you get the determinant values, put them in the formula, and you will find:

X = Dx/D

Y = Dy/D

Z = Dz/D

These are the steps to solve three equations in three variables using Cramer’s rule.

Rules for Using Cramer’s Rule in Matrix

Above, we discussed how to solve equations using Cramer's rule formula, but before using Cramer's rule in the matrix, one should know the rules to be followed to get the correct solution for the equations. Below are some rules for using Cramer’s rule in the matrix:

• While calculating the determinant value, one should note that D should never be equal to 0. If D = 0, the equation does not have any solution, or it can have infinite solutions.

• If there are n number of variables and n equations in the matrix, we must calculate the value of  (n+1) determinants.

• You can only find a solution using Cramer's rule if D is not equal to zero.