College Algebra: Evaluating 2x2 Determinants

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Okay, so we’ve been looking at multiplying matrices, seeing if two things are really compatible enough from a multiplication point of view. But now what I want to start thinking about are special matrices, matrices that are square matrices. So those are not rectangular kind of things, but in fact, they’re matrices where the number of rows equals the number of columns. Those are called square matrices, because they are sort of the shape of a square rather than sort of a long rectangle or something like this. Anyway, if you have a square matrix, you can actually associate with that collection of numbers, with that square of numbers, just one number that in some sense captures some essence of the aggregate. So the idea is you have this huge aggregate of numbers. And it’s a square matrix. So it’s like maybe 27 rows and 27 columns. You have this big thing. And what you can do is take that huge thing and associate, with that huge collection of numbers, just one number. And that one number is called the determinant. So the determinant is just a number, or a scalar if you want to think of it as a scalar. It’s just a number that you can associate with any square matrix. So how in the world do you take a whole list of numbers and take that, and make one number out? Well I’m going to show you, because it’s sort of fun.
Let’s start with just two by two matrices. So suppose I have a matrix A, and it’s just going to be two by two, so 5, 3, 8, 5. Notice that’s a square matrix, because I have two rows and two columns. So it’s a square matrix. Okay, well now if I want to compute the determinant, remember the determinant is just a number associated with this matrix, first of all let me show you what the notation is. Some people use just det(A). And that stands for the determinant of A. That stands for that special number that I’m going to tell you about in just a second. However, some people use the following notation. They just put some absolute values around it. It’s a little bit misleading in a way, because you might think it’s the absolute value of A. But it’s not. If A is a matrix, then if you see these bars around it that means determinant of A. So actually, even though I tend to use this normally in life, since I have such a small little screen here in front of you, I’m going to actually adopt this notation. But remember it’s not absolute value. It really is the determinant. It’s this special number.
How do you actually go about finding that special number? Well I’m going to tell you. With a two by two matrix, it’s actually pretty fun, because what you do is you take the product of this term with this term. So basically you take this diagonal and you multiply those numbers together. And then you subtract the product of these two things. So what you do is you take this times this, minus this times this. So it’s sort of a cross thing. If you think about it, with a two by two matrix, it’s easy to see. You just sort of take a cross, this and then minus that. So in this case, the determinant, so we use this symbol determinant of A = 5 x 5 - 3 x 8. And so, that’s 25 - 24 = 1. So in fact the determinant of this particular matrix A is 1. So 1 is the number that I associate with this determinant.
A great question to be thinking about now is what does that mean to have determinant 1? What does that mean exactly? Well we’ll see exactly what that means. And we’ll also see that, in fact, the determinant is a really important and useful tool. It really is useful, not just fake useful like we sometimes see. But first of all I want us to get a sense of how to compute these things. So before I tell you exactly the meaning of 1, let’s just make sure that we can find some of these.
Let’s take another example, B. Suppose B = 1, - 4, 2, 10. What’s the determinant of this? Well why don’t you try it right now, in fact, because you see the pattern? You take this product and subtract this product. See if you can actually find the determinant of B right now.
Okay, well let’s see. The determinant of B = 1 x 10, which is 10 minus -4 x 8, which is -8. So I have a 10 minus -8. So a negative times a negative is a positive. So, in fact, I see 8 plus 10, which 18. So this actually equals 18. So the determinant of this matrix turns out to be 18.
If the determinant of a matrix is not equal to 0, if the number we figure out is not 0, then we call the matrix nonsingular. So nonsingular just means that the determinant of that matrix isn’t 0.
One last example, C, suppose that’s 2, 6, 1, 3. Why don’t you give this a shot? See if you can find the determinant of this two by two matrix.
Well, let’s take a look. It would be 2 x 3, which is 6 - 6 x 1, which is 6. So this has a determinant actually equal to 0. So C I would not call nonsingular. In fact, what I would call C is singular. So a square matrix is singular if the determinant equals 0. If the determinant is not 0, then we say it is nonsingular.
Anyway, that’s an easy way of actually computing the two by two determinant of a matrix. Pretty straightforward, just multiply here and subtract the multiplying here.
Up next, we’ll take a look at three by three determinants, which will be a little teeny bit trickier, but as always in life, what we’re going to do is we’re going to take a harder problem, and try to reduce it to an easier problem. Because in that case, what we’re going to see is, for example, a matrix like this, a three by three, three rows, three columns. We just can’t do that cross thing, because there are a lot of terms here. So what we have to do is figure out a way of getting around that. And the way we’re going to do it is by dividing and conquering. We’re going to sort of break this up and just see little two by two matrices, and then compute the determinants of those, and string them together, or knit them together, if you are so inclined.
Anyway, up next we’ll talk about that, see how to do that, and get a sense later of the value of determinants, and how you can use them. You’ll be amazed. A lot of the algebra we’ve been doing with simultaneous equations can be just completely avoided. And in fact, you can actually solve big systems of equations just using determinants. Wow. I’ll see you soon.