Using Cramer's Rule with Inconsistent and Dependent Systems

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  • 0:01 Cramer's Rule
  • 0:56 Inconsistent and…
  • 1:20 The Coefficient Determinant
  • 3:01 Can We Use Cramer's Rule?
  • 3:24 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson, and you will see what happens when we use Cramer's Rule with inconsistent and dependent systems. You will see what kind of result you will always get when you try to use Cramer's Rule.

Cramer's Rule

In this lesson, we talk about using Cramer's rule. Cramer's rule is a very neat way to solve linear systems for its various variables. You can solve for just one variable using Cramer's rule without having to solve the whole system. What you do is create a matrix from the coefficients. Then you find the determinant of this matrix. We will call this determinant D.

Next, say we want to find the y value of our system, we would then replace the y column of our coefficient matrix with the constant values on the right side of our equations. We find this determinant and call it D sub y. We then divide our D sub y with our D, and we end up with our y solution. It's pretty neat.

Inconsistent and Dependent Systems

So, we know we can use Cramer's rule with linear systems that have a unique solution. But what about using it with inconsistent systems, systems with no solution, and dependent systems, systems with infinite solutions? What happens when we use Cramer's rule in these instances? Let's find out.

The Coefficient Determinant

Let's use Cramer's rule to try and solve this linear system. I will tell you that this system is a dependent system with an infinite number of solutions.


According to Cramer's rule, we first create our coefficient matrix. Our first row in this matrix is 1, 1, and 1. Our second row is 2, 2, and 2. Our third row is -1, -1, and -1. We then take the determinant of this matrix.


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