Cramer's Rule is used to find the unknowns in the given system of linear equations. Cramer's Rule is the most commonly used formula for finding the solution to the given system of linear equations in matrix form.
- This rule is valid only if the given system of equations has a unique solution.
- It doesn't work with a system of equations with infinitely many solutions or no solution.
- This rule is used to find solutions for variables with the same number of equations.
- This rule uses determinants to find the solution of the given equations or the value of unknowns.
Formula
Cramer's Rule formula is used to solve the system of equations in the form
AX = B
- A is the coefficient matrix
- B is the column matrix of constants
- X is the column matrix of unknowns
Now the value of X is calculated using the formula given in the diagram below,
The individual values of x, y, and z are calculated with the help of the formula discussed in the above figure.
Conditions
Cramer’s rule is applicable only when certain conditions are satisfied. The important condition of Cramer's rules are,
- The system AX = B has a unique solution if D ≠ 0 i.e. determinant is non-zero.
- If D = 0 we have two conditions and any of them can be true,
First Condition: Infinitely many solutions, this situation arises when D = 0 and at least one determinant of the numerator is zero.
Second Condition: No solution, this situation arises when D = 0 and at no determinant of the numerator is zero.
Cramer's Rule For 2 x 2
Now let us solve a system of 2 equations in 2 variables using Cramer's rule. Given equation,
Follow the steps to solve the system of 2 × 2 equations with two unknowns x and y using Cramer's rule.
Step 1: Write the given system of the equation in matrix form as AX = B
Step 2: Find the determinant (D) of A and find Dₓ and Dᵧ where
Dₓ = det (A) where B replaces the first column of A
Dᵧ = det (A) where B replaces the second column of A
Step 3: Find the values of the variables x and y as,
- x = Dx / D
- y = Dy / D
Cramer's Rule For 3 × 3
Now let us solve a system of 3 equations in 3 variables using Cramer's rule. Given equation,
- a1x + b1y c1z = d1
- a2x + b2y c2z = d2
- a3x + b3y c3z = d3
Follow the steps to solve the system of 3 × 3 equations with two unknowns x and y using Cramer's rule.
Step 1: Write the given system of the equation in matrix form as AX = B
Step 2: Find the determinant (D) of A and find Dx, Dy, and Dz where
- Dx = det (A) where B replaces the first column of A
- Dy = det (A) where B replaces the second column of A
- Dz = det (A) where B replaces the third column of A
Step 3: Find the values of the variables x, y, and z as,
- x = Dx / D
- y = Dy / D
- z = Dz / D
Steps to Solve Using Cramer's Rule
Study the following steps to solve the linear equations using Creamer's Rule,
- Write the given system of equations in AX = B form.
- Find the value of determinant (D) of matrix A. (Note: If the determinant is zero, then a system of equations does not have a unique solution, which is invalid in Cramer's Rule).
- Now, find the value Dx which is the determinant of matrix A in which constants of the given linear equations replace the coefficient of x.
- Now, find the value Dy which is the determinant of matrix A in which constants of the given linear equations replace the coefficient of y.
- Now, find the value Dz which is the determinant of matrix A in which constants of the given linear equations replace the coefficient of z. (find this determinant only if 3 variables are present in the given equation).
- Similarly, find determinants for all the unknowns if more than three unknowns are present.
- Find the values of x = Dx/D, y = Dy/D, and z = Dz/D.
➢Practice: Solved Examples
