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Cramer’s Rule | Cramer’s Rule Calculator 2 & 3 Equations System

Cramer’s Rule & Calculator for Linear Circuit Analysis | Step by Step with Solved Examples

Today, we are going to share another simple but powerful circuit analysis technique which is known as “Cramer’s Rule“.
Update: We have added Online Cramer’s Rule Calculator where you can solve two equations system as well as three equations system. Check both Cramer’s rule calculator in both sections of the post. Thanks
 

Cramer's Rule | Cramer's Rule Calculator 2 & 3 Equations System

Below is the Step by Step tutorial of solved examples, which elaborates that how to solve a complex electric circuit and network by Cramer’s rule.

Cramer’s Rule Calculator for 2×2 (Two Equations System)

Cramer’s Rule Calculator 2 x 2 (2 Equations System):
x + y =
x + y =
x =
y =

Finding Two Variables by Cramer’s Rule:

Example 1:
(In our case, the unknown values are two currents which are i1 and i2) by Cramer’s Rule. Now, let’s begin.
 
As shown below, this is a simple electric circuit and we are going to solve it by Cramer’s rule.
 Cramer's rule. step by step procedure with solved example
Cramer’s Rule for Linear Circuit Analysis | 2 Variables (2×2) Solved Example.
Solution:
First, rearrange the circuit with proper labels (As two of 5Ω resistors are in series, so, we will replace it with 10Ω.
Apply Mesh Analysis and Simplify by Cramer;s Rule to find the unknown values of Currents.png
Apply Mesh Analysis and Simplify by Cramer’s Rule to find the unknown values of Currents i1 and i2.
Now, we will write the KVL equations of unknown values for the given circuit
Apply KVL on Mesh (1).
6 = 14i1 + 10(i1i2)
                6 = 24i1 – 10i2 ….. → Eq (1)
Also, Apply KVL on Mesh (2).
-5 = 10 i2 + 10(i2 i1)
                   -5 = – 10 i1 + 20 i2 ….. → Eq (2)
Here, we got two equations, i.e.
     24 i1 – 10 i2  =   6
  – 10 i1 + 20 i2 = -5
 
Now, we will solve these two equations by Cramer’s rule to find the unknown values (of currents), which are i1 and i2.
 
Solving by Cramer’s rule:
Step 1:
First of all, write the above equations in the matrix form. i.e.cramer's rule. step 1 matrix form
Step 2:
Now, write the coefficient matrix of the above equations and call it ∆. Make sure it is square, i.e. Number of Rows x Number of Columns. In the above case, it has 2 rows and 2 columns.cramer's rule for electrical engineering
 
Step 3:
Now find the determinant |∆| of the coefficient matrix ∆ by the following method. (The given fig below will help you to do so.)
Click image to enlarge
Cramer's rule. step by step procedure with solved examples of two and three variables
Finding coefficient matrix of ∆ for Cramer’s Rule. Easy Explanation.
According the above fig. the final step would be like this.Cramer's rule. easiest method ever to solve equations
Step 4:
Now find the coefficient determinant of Δ1 by the same method as mentioned above, but replace the first column of Δ with the “Answer column” (If you didn’t get the point of the answer column, See the fig in step 2 above or check the infographics at the end of the example of just refer to the second example below, where we did the same to find Δ1) which is,cramer's rule
Step 5:
Now find the coefficient determinant of Δ2, just replace the second column with the “Answer Column” which is,cramer's rule. easy explanation.png
Step 6:
As Cramer’s rule tells that i1= Δ1 / Δ and i2 = Δ2 / Δ.
Now, Find i1 and i2 by Cramer’s rule.
cramer's rule 2x2
i1= 0.184.2 A or 184.2 mA
 
And,
cramer's rule formula
i2 = 0.157.9 A or 157.9 mA
 
Below is the infographics summary of Cramer’s rule to determine the two variables or unknown values.
Cramer's Rule Simple Steps Infographics Chart
Cramer’s Rule Simple Steps Infographics Chart
Ok, it was easy… Now, How about 3 variables…. Let’s try to solve 3 variables linear equations with the help of Cramer’s rule.

Finding Three Variables by Cramer’s Rule:

(In our case, these unknown values are three currents which are i1, i2 and i3) by Cramer’s Rule. Now, let’s begin.

Cramer’s Rule Calculator for 3×3 (Three Equations System)

Cramer’s Rule Calculator 3 x 3 (3 Equations Systems ):
x + y + z =
x + y + z =
x + y + z =
x =
y =
z =
 
Example 2:
Use Mesh Analysis to determine the three mesh currents in the circuit below. Use Cramer’s rule for simplification.
Solving Linear Electric circuit of 3 equations by Cramer's Rule. Example
Find the three unknown Values of Currents by Cramer’s Rule.
First of all, apply the KVL on each mesh one by one, and write its equations.
            -7+1(i1i2) +6+2(i1i3) = 0
         1(i2 i1) + 2i2 + 3(i2 i3) = 0
      2(i3 i1) – 6+3(i3 i2) +1i3 = 0
Simplifying,
     3i1 i2 – 2i3 = 1                    … Eq….. (1)
  – i1 + 6i2 – 3i3 = 0                    … Eq….. (2)
 -2i1 – 3i2 + 6i3 = 6                    … Eq….. (3)
 
Now, write the above equations in the matrix form.
     3i1i2– 2i3 = 1
   –i1+ 6i2– 3i3 = 0
-2i1– 3i2+ 6i3 = 6
 
solving three equations sytem by cramer's rule 
Now, we will find the coefficient determinant of ∆. How will we do that? Just check the fig below for better explanation.
Click image to enlargeCramer's rule. step by step procedure with solved examples of two and three variables
 So the full step is shown below.3x3 cramer rule
∆ = +3 (6 x 6) – (- 3 x –3) – (-1 (-1 x 6)-(-2 x –3) + (-2 (-1 x –3) – (-2 x 6)
= 81 -12 -30 = 39
 
Now, find the ∆1 by the same way as explained above. But, just replace the first column of the matrix with the “Answer Column”. For detail, check the fig shown below.cramer's rule. easy explanation
So, here is the full step to find ∆1. Here, we replaced the “Blue Guys” in the first column with “Black Guys” :).solving electric circuits by cramer rule
= +1(36-9) – (1[0+18]) –2(0-36)
= 27 + 18 + 72
1 = 117
 
Again, find the ∆2 with the same method as explained earlier. Just replace the second column of the matrix with the “Answer column” i.e. replace the “Red guys” in the center column with “Black Guys” as shown below.solve linear circuits by cramer's rule
= +3 (0 +18) -1[(-6)-(+6)] –2(-6-0)
= 54+12+12 = 78
2 = 78
 
Finally, find the last ∆3. Just replace the third column with the “Answer column” i.e. replace the “Green guys in the third column with “Black guys” as shown below.cramer rule for circuit analyzing and simplification
= +3 (6 x 6) – (-3 x 0) – [-1(-1 x 6) – (-2 x 0)] + [1(-1) x (-3) – (-2) x (6)]
= 108 + 6 + 15
3 = 117
 
Now, solve and find the unknown values of current, i.e. i1, i2 and i3.
As, Cramer’s rule says that, variables i.e. i1 = ∆1/∆1, i2 = ∆/∆2 and i3 = ∆/∆3.
Therefore,
 
i1 = ∆1/∆1
= 117/39
i1 = 3A
 
And i2,
i2 = = ∆/∆2
= 78/39
i2 = 2A
 
And finally, i3;
i3 = ∆/∆3
= 117/39
i3 = 3A.
 
I hope that you understood the cramer’s rule very well and enjoyed the step by step tutorial. Please, don’t forget to share with your friends. Also, enter your email address in the below box to subscribe. So, we will send you more tutorials like the above one. Thanks.

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One comment

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