Intro to determinant notation and computation | Matrices | Precalculus | Khan Academy

In this video, we're going to 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 2 by 2 matrix here. So, and actually I'll give it in general terms. 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 let me do that in a different color, so this is D right over here.

The determinant of this matrix. So let's actually let me just call this matrix, let's say that this is matrix A. So there's a bunch of ways to say, 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, 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 times the bottom right, so A times D, minus the top right times the bottom left, B C. 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 1, negative 2, 3, and 5. 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 1 times 5 minus the product of these two numbers, which is 3 times negative 2. And that, of course, is going to be equal to 1 times 5 is 5, 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 the future video I will give you an interesting interpretation of the determinant.