Cramers Rule

Cramer's rule is invented by mathematician Gabriel Cramer in 1750s. This rule is used to find the solution of a system of equations with any number of variables and the same number of equations. Sometimes, when we are solving a system of equations in 3 variables, say x, y, and z, we may need to solve for two variables x and y to solve for variable z. But using Cramer's rule, we can find the value of any variable without finding the values of the other variables.

But this rule has some limitations with respect to the solutions. This rule can be applied only when the system has unique solutions. But how do we know when a system has unique solution? Let us learn more about this along with the definition and formula of Cramer's Rule.

Cramer's rule is one of the methods used to solve a system of equations. This rule involves determinants. i.e., the values of the variables in the system are found with the help of determinants. Let us consider a system of equations in n variables x₁, x₂, x₃, ..., xₙ written in the matrix form AX = B, where

• A = the coefficient matrix which is a square matrix
• X = the column matrix with variables
• B = the column matrix with the constants (which are on the right side of the equations)

Here is the Cramer's rule formula to solve the system AX = B (or) to find the values of the variables x₁, x₂, x₃, ..., xₙ. To solve the system of equations:

• Find det |A| and represent it by D.
• Find the determinants Dₓ₁, Dₓ₂, Dₓ₃, ..., Dₓₙ, where Dₓᵢ is the determinant of matrix A where the i^th column is replaced by the column matrix B. 
• We divide each of these determinants by D to find the value of the corresponding variables. i.e.,
  x₁ = Dₓ₁/D, x₂ = Dₓ₂/D, ...., xₙ = Dₓₙ/D.

Note that the system of equations has a unique solution only when D ≠ 0.

Are you getting confused with this general formula of Cramer's rule? Let us see this rule for 2 x 2 and 3 x 3 system of equations for clarification.

Using the above formula, let us see how to solve a system of 2 equations in 2 variables using Cramer's rule. Here are the steps to solve this system of 2x2 equations in two unknowns x and y using Cramer's rule.

• Step-1: Write this system in matrix form is AX = B.
• Step-2: Find D which is the determinant of A. Also, find the determinants Dₓ and Dᵧ where
  Dₓ = det (A) where the first column is replaced with B
  Dᵧ = det (A) where the second column is replaced with B
• Step-3: Find the values of the variables x and y by dividing each of Dₓ and Dᵧ by D respectively.

Consider a system of two equations in two variables x and y.

a₁x + b₁y = c₁ and

a₂x + b₂y = c₂

Let us apply the above steps to solve the above system.

Step-1: Write this system in matrix form is AX = B, where

• A = \(\left[\begin{array}{ll}a_{1} & b_{1} \\ \\ a_{2} & b_{2}\end{array}\right]\) = the coefficient matrix
• X = \(\left[\begin{array}{l}x \\ \\ y\end{array}\right]\) = the variable matrix
• B = \(\left[\begin{array}{l}c_1 \\ \\ c_2\end{array}\right]\) = the constant matrix

Step-2: Calculate the determinants D, Dₓ, and Dᵧ, where 

• D = det (A) = \(\left|\begin{array}{ll}a_{1} & b_{1} \\ \\ a_{2} & b_{2}\end{array}\right|\)
• Dₓ = det (A) where the first column is replaced with B = \(\left|\begin{array}{ll}c_{1} & b_{1} \\ \\ c_{2} & b_{2}\end{array}\right|\)
• Dᵧ = det (A) where the second column is replaced with B = \(\left|\begin{array}{ll}a_{1} & c_{1} \\ \\ a_{2} & c_{2}\end{array}\right|\)

Step-3: Find x and y (when D ≠ 0) using

• x = Dₓ/D
• y = Dᵧ/D

We will just extend the same process of Cramer's rule for 2 equations for a 3x3 system of equations as well. Here are the steps to solve this system of 3x3 equations in three variables x, y, and z by applying Cramer's rule.

• Step-1: Write this system in matrix form is AX = B.

• Step-2: Find D which is the determinant of A. i.e., D = det (A). Also, find the determinants Dₓ, Dᵧ, and D_z where
  Dₓ = det (A) where the first column is replaced with B 
  Dᵧ = det (A) where the second column is replaced with B
  D_z = det (A) where the third column is replaced with B

• Step-3: Find the values of the variables x, y, and z by dividing each of Dₓ, Dᵧ, and D_z by D respectively.

Consider a system of three equations in three variables x, y, and z.

a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂ and

a₃x + b₃y + c₃z = d₃

Let us apply the above steps to solve 3x3 equations.

Step-1: We will write the system in matrix form is AX = B, where

• A = \(\left[\begin{array}{ll}a_{1} & b_{1} &c_1\\ a_{2} & b_{2}&c_2\\a_3&b_3&c_3\end{array}\right]\) = the coefficient matrix
• X = \(\left[\begin{array}{l}x \\ y \\z\end{array}\right]\) = the variable matrix
• B = \(\left[\begin{array}{l}d_1 \\ d_2 \\d_3\end{array}\right]\) = the constant matrix

Step-2: Compute the determinants D, Dₓ, Dᵧ, and D_z where

• D = det (A) = \(\left|\begin{array}{ll}a_{1} & b_{1} &c_1\\ a_{2} & b_{2}&c_2\\a_3&b_3&c_3\end{array}\right|\)
• Dₓ = det (A) where the first column is replaced with B = \(\left|\begin{array}{ll}d_{1} & b_{1} &c_1\\ d_{2} & b_{2}&c_2\\d_3&b_3&c_3\end{array}\right|\)
• Dᵧ = det (A) where the second column is replaced with B = \(\left|\begin{array}{ll}a_{1} & d_{1} &c_1\\ a_{2} & d_{2}&c_2\\a_3&d_3&c_3\end{array}\right|\)
• D_z = det (A) where the third column is replaced with B = \(\left|\begin{array}{ll}a_{1} & b_{1} &d_1\\ a_{2} & b_{2}&d_2\\a_3&b_3&d_3\end{array}\right|\)

Step-3: Find the values of the variables x, y, and z (when D ≠ 0) using

• x = Dₓ/D
• y = Dᵧ/D
• z = D_z/D

If we observe the formula of Cramer's rule in all the above three sections, we have mentioned that D ≠ 0 everywhere. This is because while finding the values of the variables, D is in the denominator and if D = 0, the fraction (the value of the variable) goes undefined. So this rule is applicable only when D ≠ 0. But what about the system of equations when D = 0? Then there are two possibilities.

• The system may have no solution.
• The system may have an infinite number of solutions.

Though Cramer's rule doesn't help in finding the infinite number of solutions, we can determine whether the system has "no solution" or "infinite number of solutions" using the determinants which we compute as the process of applying the rule.

• If D ≠ 0, we say that the system AX = B has unique solution.
• If D = 0 and atleast one of the numerator determinants is a 0, then the system has infinitely many solutions.
• If D = 0 and none of the numerator determinants is 0, then the system has no solution.

You can visualize this from the following Cramer's rule chart.

From the above chart and explanation, it is very clear that Cramer's rule is NOT applicable when D = 0. i.e., when the determinant of the coefficient matrix is 0, we cannot find the solution of the system of equations using Cramer's rule. In this case, we can find the solution (if any) by using Gauss Jordan Method.

Thus, Cramer's rule is used to find the solution of a system only when the system has a unique solution.

Important Notes on Cramer's Rule:

Here are some important notes related to the application of Cramer's rule:

• If there are n variables and n equations, we have to compute (n + 1) determinants.
• This rule can give the solutions only when D ≠ 0.
• If D = 0, the system has either an infinite number of solutions or no solutions.
• We cannot find solutions by using this rule when the system has an infinite number of solutions.

☛Related Topics:

• Cramer's Rule Calculator
• Linear Equations in Two Variables
• Graphical Approach in Linear Equations
• Simultaneous Linear Equations

How Does Cramer's Rule Work?

Cramer's rule is used to find the solution of the system of equations with a unique solution. It is also used to find whether the system has a unique solution, no solution, or an infinite number of solutions.

What is Cramer's Rule Definition?

Cramer's rule says the solution of the system of equations written in the matrix form AX = B (where A is the matrix of coefficients, X is the column matrix of variables, and B is the column matrix of coefficients) is obtained by dividing det (A) by the same determinant where the respective columns are replaced by the matrix B.

Who Invented Cramer's Rule of Matrices?

Cramer's rule for matrices is invented by a mathematician called Gabriel Cramer in 1750. He is a Swiss mathematician. This rule is very helpful in finding any variable right away without needing of finding any other variables.

What is Cramer's Rule 2x2?

First, write the given system of 2x2 equations as AX = B, where X is a column matrix of the variables x and y. Then find the determinants D, Dₓ, and Dᵧ, where D = det(A) and Dₓ and Dᵧ are same as det(A) where the first and second columns are respectively replaced by the matrix B. Then use the following to find the variables x and y.

• x = Dₓ/D
• y = Dᵧ/D

How Do You Use Cramer's Rule for 2x3 Equations?

Cramer's rule deals with the determinants and determinants can be found only for square matrices. But if we write 2x3 equations in the form of AX = B, then A is NOT a square matrix (it is a rectangular matrix) and hence this rule cannot be applied in this case.

What is Dₓ in Cramer's Rule?

To solve a system of equations using Cramer's Rule, first, we write it in the form AX = B. Then Dₓ is a Cramer's rule determinant of the coefficient matrix where the first column is replaced with the column matrix B.

What are the Advantages of Cramer's Rule?

Cramer's rule is used to solve the system of equations where the number of variables is equal to the number of equations. Also, using this rule, we can find the value of finding any variable right away without finding the other variables.

How to Apply Cramer's Rule When the Determinant is Zero?

While solving a system AX = B using Cramer's rule, if det A = 0, then the system either has an infinite number of solutions or no solution. In either case, we cannot conclude/find anything using Cramer's rule. This is because while finding every variable using Cramer's rule formula, we have to divide the determinants by det A and a fraction is undefined when its denominator is 0. Hence, when det A = 0, the Cramer's rule cannot be used.

What is Cramer's Rule 3x3?

First, write the given system of 3x3 equations as AX = B, where X is a column matrix of the variables x, y, and z. Then find the determinants D, Dₓ, Dᵧ, and D_z, where D = det(A) and Dₓ, Dᵧ, and D_z are same as det(A) where the first, second, and third columns are respectively replaced by the matrix B. Then use the following to find the variables x, y, and z.

• x = Dₓ/D
• y = Dᵧ/D
• z = D_z/D