Intro to determinant notation and computation
About this video
This video introduces the determinant of a 2-by-2 matrix. It shows several ways to write it, including det(A) and vertical bars, and defines it as ad minus bc. It then works an example, finding that the determinant of the matrix with rows 1, -2 and 3, 5 is 11.
You'll be able to
- recognise the different notations for the determinant of a matrix.
- compute the determinant of a 2-by-2 matrix using ad - bc.
- handle negative entries correctly when computing a determinant.
Key words
- Determinant
- A single number computed from a square matrix; for a 2-by-2 matrix with rows a, b and c, d it is ad - bc.
- det(A)
- A notation meaning 'the determinant of matrix A'.
- Vertical bars |A|
- Bars around a matrix mean its determinant, not absolute value.
Transcript
- [Instructor] In this video, we're gonna talk about something called determinants of matrices. So I'll start just telling you the notation and how do you compute it. And then we'll think about ways that you can interpret it. So let's give ourselves a two by two matrix here.
So, and actually, I'll give it in general terms. So let's say that this top-left term here is A, and then this one here is B the top-right. The bottom-left is C, and then let's call this bottom-right D. And I'm gonna do that in a different color. So this is D right over here.
the determinant of this matrix. And actually, let me just call this matrix. Let's say that this is matrix A. So there's a bunch of ways to call the determinant, or have the notation for the determinant. We could write it like this. We could have these little, it looks like absolute value signs, but it really means determinant when you apply it to a matrix.
So the determinant of matrix A. You can write it that way. You could write it this way, the determinant of matrix A. You could write it that way. Or you could write it this way, where you put these lines that look like big absolute value signs instead of the brackets when you describe the numbers.
So you could also write it this way. And I haven't explained what determinant is or even how to compute it yet. I'm just talking about the notation of how you even talk about the determinant of a matrix. So you can also write it this way, just rewrite the whole matrix with those vertical bars next to it.
This is defined as, and we'll see how it's useful in the future, the top-left time's the bottom-right. So A times D minus the top-right times the bottom-left. BC. So another way to think about it, it is just these two, the product of these two minus, so that's those two right over there, minus the product of these two right over here.
So let's just first, before we start to interpret this, get a little practice, just computing a determinant. So let me give you a matrix. So let's say I have the matrix one, negative two, three, and five. Pause this video and see if you can compute the determinant of this matrix.
Let's call this matrix B. I want you to figure out the determinant of matrix B. What is this going to be equal to? All right, now let's do this together. So you're going to have the product of these two numbers. So we have one times five minus the product of these two numbers, which is three times negative two.
And that of course, is going to be equal to one times five is five, three times negative two is negative six. But we're subtracting a negative six. Five minus negative six is the same thing as five plus six which is going to be equal to 11. Now that we know how to compute a determinant, in a future video, I will give you an interesting interpretation of the determinant.
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