Hey, everyone. Welcome back. So, throughout our discussion on matrices, one of the things that you may be asked to calculate is something called the determinant of a matrix. Now, the first time you learn about determinants, it may be kinda scary because there's lots of letters and numbers flying around everywhere, but I'm going to break it down for you and show you that the determinant is really just a number that you calculate by doing some subtraction and multiplication. So, I'm going to show you that the determinant of this matrix over here is actually just the number 2. I'm going to break it down for you and show you exactly how we get that. Now later on, we're going to use this determinant to solve systems of equations, but for now, we only need to worry about how to calculate it. Let's go ahead and get started here.

We're going to look at a 2 by 2 matrix because the formula depends on the type of matrix, and this is the simplest version. The way we calculate this is basically by subtracting the products of the diagonals. Now that sounds like a lot, but let me just break it down for you. Right? So we've got this matrix here, A, which is 3, 2, 5, and 4. So we've got columns and rows. Right? This is the rows, and this is the columns. But the diagonals are really just like the opposite corners of your matrix. So the diagonals here are just basically these two numbers over here and then these two numbers over here. So the way that you calculate this matrix, and by the way, you're going to see some different notation for this, like det A or sometimes even these little straight bars, is you're going to multiply these diagonals. You're going to multiply these two numbers. You always do the blue ones first, and then you go do these two numbers. Alright? So you always go down to the right and then down to the left, and you subtract those. Alright? So really what happens is this just becomes a subtraction of 2 products. So I've got that's the first two, our numbers that multiplied, are going to be 3 and 4. So this is going to be 3 times 4 minus you're going to subtract, and then 2 times 5. You always do the blue ones first, the red ones second. Alright? So this is going to be 2 times 5. It doesn't matter which way you multiply these because the answer is the same. And so what this ends up being is this ends up being 12 minus, and this ends up being 10. And that's why we got our answer of 2. So the determinant of this matrix, which is really just a number, is just the number 2. Alright? So that's really all there is to it. In fact, the general formula for doing this, for the determinant of some 2 by 2 matrix, is if you have these numbers arranged in like a grid, then you again always do these two numbers. So in other words, a×d−b×c. That's the general formula for any determinant of a 2 by 2 matrix. Alright? So hopefully, you guys remember that. That's really all there is to it. So let's go ahead and just get some more practice here.

We're going to evaluate the determinants of these matrices. So here we have matrix A, which is given to us as 8, 4, 5, and 0. So the way we would write this is, again, the determinant of A, or you could also just use the square bracket notation where there are straight lines. The determinant of this is we're going to take a look at again, we're going to subtract these two products over here. Remember, these numbers will go first. So in other words, we're going to do 8 times 0, and then we're going to do minus 4 times 5. Alright? Now one of the things you might notice here is that if you do 8 times 0, that just ends up being 0. So this is just 0 minus, and then this over here is just going to be 20. Alright? So in other words, the determinant of this matrix is just the number negative 20. Alright? That's really all there is to it. Just really nothing else to sort of read into in terms of what does this number mean. It's just a number, and we'll use it later on. Alright?

So, let's take a look at the second matrix. We have B, and here we have some negative numbers. Instead of using the det A notation, you also may see this written like this. You may see this written as just these little straight lines with B. And, again, this is just going to be the subtraction of 2 products. Alright? So, again, you're going to go down to the right and then down to the left, and you're going to do this multiplication in that order. Alright? So this is going to be negative 3 times negative 2, and then you're going to subtract negative 7 times 1. Alright? So be careful with the negative signs here. What we're going to see here is that this first product ends up being negative 3 times negative 2, which actually just works out to just positive 6 because the negatives cancel. So this is just positive 6. And then we have minus, and this is just going to be negative 7. Alright? So this is going to be negative 7. So here we have 6 minus negative 7. And so, in other words, the determinant of B is just going to be the number 13. Alright? So that's your answer. That's all there is to it, folks. Let me know if you have any questions. Thanks for watching.