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Determinant: Definition & Meaning

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

The determinant of a matrix is a useful tool that helps you 'determine' whether or not a system of equations has a unique solution. In this lesson, you will learn about determinants and their uses.

Definition

The determinant of a matrix is simply a useful tool. Like its name suggests, it 'determines' things. When doing matrix algebra, or linear algebra, the determinant allows you to determine whether a system of equations has a unique solution.

Unique Solutions and the Determinant

A system of equations is simply a set of more than one equation with two, three, or more variables. Suppose, for example, that you have the following system of equations:

x + y = 3 (Equation A)

2x + y = 4 (Equation B)

In Equation A, there are many combinations of x and y that will work. For example, x = 3 and y = 0 is a solution, but so is x = 0 and y = 3. In fact, there are an infinite number of solutions because fractions and decimal numbers are fair game. x = 1.22 and y = 1.78 would work.

Similarly, there are an infinite number of solutions to Equation B, but most of them would not also be solutions for Equation A. In this case, only one solution to Equation A, x = 1 and y = 2 also works for Equation B. There is only one solution to the system of equations - only one solution that works for both equations. So, the system of equations has a 'unique solution.' There is only one x-y pair that works. If you graph both equations, you will see that they cross at a single point.

Equations that intersect at one point
Graph of two intersection lines

Suppose, instead, that you have the following two equations:

x + y = 5 (Equation C)

x + y = -1 (Equation D)

Okay, think about that for a minute. How can you add two things together and get '5' and add the same two things together and get '-1'. Of course, you can't. The two equations are inconsistent. There is no x-y pair that will work in both equations simultaneously. There is no solution to the system of equations. If you graph the two lines, they never cross (they are parallel).

Parallel lines never cross
Graph of Parallel Lines

Alternatively, you might have the following two equations:

x + y = 5 (Equation E)

2x + 2y = 10 (Equation F)

In this case, if you look closely, you will see that Equation F is just Equation E with everything multiplied by two. The two equations represent the same relationship between x and y. x = 1 and y = 4 works in both equations, but you would find that every solution that works for Equation E also works for Equation F. If you graph the two equations, you would find they fall right on top of each other. They are the same.

Two lines that are the same
Lines that are the same line

In fact, you could simplify Equation F, and you would get a system that looks like this:

x + y = 5 (Equation E)

x + y = 5 (Equation F)

If you compare Equations C and D with Equations E and F, you will see something interesting - the left hand sides of the equations all look the same. In C and D, the right hand sides are different from each other. In E and F, the right hand sides are identical.

The pattern on the left side, however, gives you important information. If you didn't have the right hand sides, just by looking at the left side, you would know something important: the two lines are either parallel (they don't cross) or they are the same line. You might not know which but you know it is one of those two. The two equations do not meet in a point. There is not a unique solution (just one crossing point) to the system of equations.

In case you are wondering how you look at those left hand sides to determine whether or not there is going to be a unique solution, don't worry. That is coming. The secret is in looking at the pattern of the coefficients, the numbers in front of the x and y for each equation. By looking at that pattern alone, you can determine whether or not the system will have a unique solution. And, yes, there is a reason why determine has been emphasized with italics in this paragraph. The pattern of coefficients relates to determining something - from those coefficients you get the determinant.

The determinant of a matrix is simply a mathematical toy that helps you know whether a system of equations will have a unique solution. You get it by playing around with the coefficients on the left hand side.

How to Find a Determinant

Finding a determinant can be tricky, especially if you have a large system of equations - lots of variables and lots of equations. In this short lesson, we will just show you how to find the determinant of a 2x2 matrix, meaning a situation in which you have only two equations and two variables.

For example, from Equation A and Equation B, you can create a matrix using the coefficients on the left hand side:

A Matrix

If you rewrite the matrix using perfectly straight lines on the outside edges, instead of brackets that bend inward, it means to find the determinant.

Determinant of a matrix

If given the matrix a name, such as matrix 'D', then you can also indicate that you wish to find the matrix by putting straight lines on both sides of the name of the matrix like this: |D|. Or sometimes it is written this way: det(D).

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