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.
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.
Taking the determinant, we get 1(-2-(-2)) - 1(-2 - (-2)) + 1(-2 - (-2)) = 1(0) - 1(0) + 1(0) = 0. Look at that. We got 0 for our determinant D. If we wanted to find the x solution, Cramer's rule tells us that we need to replace our x column with the constant numbers. So, we would replace our first column with the numbers 1, 2, 1 from the right side of our equations. We then find this determinant and divide it by our coefficient determinant D. But wait, our coefficient determinant D is 0. We can't divide by 0.
What does this tell us?
Can We Use Cramer's Rule?
This tells us that Cramer's rule can tell us that our linear system is either inconsistent or dependent, but it can't tell us which it is. So, can we use Cramer's rule in these instances? No, we can't, since division by 0 is not a valid math operation.
So, what have we learned? We've learned that Cramer's rule is a very neat way to solve linear systems for its various variables. Inconsistent systems are systems with no solutions. Dependent systems are systems with an infinite number of solutions. We can't use Cramer's rule with these systems because we get a coefficient determinant of 0. When we get a coefficient determinant of 0, we can't continue with Cramer's rule because it would result in division by 0.
Following this lesson, you'll be able to:
- Define inconsistent systems and dependent systems
- Explain why Cramer's rule cannot be used for these types of systems