We learned what a determinant is. We sought an action with a two-by-two matrix. Let's see it in action with a three-by-three matrix. Here, we want to do with the three-by-three matrix, and let's call it A, and we want to be dealing with the determinant of A, which again can be written as two vertical lines and two vertical bars outside of the letter of the matrix. Let's setup a hypothetical, a, b, c, d, e, f, g, h, and i. Now a three-by-three matrix gets a little bit more complicated. There is a very procedural way of doing a three-by-three matrix or above, and this strategy can expand pretty much infinitely to any size, but we'll start with the three-by-three, and I'll talk about doing a four by four or five by five, but those get pretty work-intensive. It might take you a long time. It's susceptible to errors. At the point where you're doing the determinant of a four by four or five-by-five, it might just be more optimal for you to plug it into a determinant calculator or an inverse calculator if you can at this point. But learning how to do it with a two-by-two, it's definitely important for the foundation, and a three-by-three, it's definitely important for the foundation. Then from there, you can expand it to anything bigger. If you have to do it, you'll know how to do it. This general strategy with three by three matrix comes with a very quirky way of figuring it out. The general strategy of what you want to do, is you want to pick one of the three rows, and you want to take one of the rows and that'll be your starting row. The best way of learning it, it's to just simply talk about how to do it. Let's say I pick the first row. There is always a plus minus coefficient matrix that attaches itself to this, so for a three-by-three, and it's always alternating, every row and column from there will be alternating. The top spot is always plus. We're going to do an example of both of these because it can be confusing to just see this and not see an example of this. The coefficient plus-minus, we'll start with a plus, and then we go minus plus. From here, it's a plus. Here, we go minus plus. Then every other one. Pretty much vertically and horizontally, no minus should be touching another minus, no plus should be touching another plus. This will talk about when you pick a certain row, what positive or negative is being attached to your starting values. Here it's saying that if I were to pick the first row, my a would be positive. Then when I go to the next one, my b would be negative. Then when I go to the next one, my c would be positive. Let's say I started with the second row. Again, you can start with any row you want when you calculate these determinants. If I started with the second row, my d would be negative, my e would be positive, my f would be negative. It's corresponding to this exact thing. Now, if I had a four by four matrix, it would be the same thing: plus, minus, plus, minus, minus, plus, minus, and you would just fill this out as you go alternating everyone, but we're only really dealing with a three-by-three for this foundational example. When we do the next two examples, which is the same matrix, but we're starting with a different row, we're going to use this information. The general strategy here, is I'm going to choose a row. Again, I'm going to choose the first row, and I have a plus, so I have a plus a, so just whatever a is, I'll have there, and then I'll have the determinant of whatever is left when I get rid of the row and column that's touching a. For a, it'll be this column, and this row will go away. For a, it will be this, that's leftover. Because once I choose my a, I'm crossing out the row associated with a and the column associated with a. This column and this row, go away completely, and I'm left with the determinant of whatever's left, e, f, h, and i. That's only the first part. I'm done with a. Now I chose the first row. That's why my a is here. Now I go along my row, I say I'm going to have a b. Is it positive or negative? I go to my little chart here, and it's negative. It's actually minus b determinant of whatever is left when I get rid of the row and column associated with b, so b is in the first row. I get rid of that. Let me get a different color here, so a was this color. Let's see if I can do different colors here. B will be blue. Again, when I get rid of the column that b is in and the row that b is in, I'm left with this and this. That's what I'm going to bring over, d here, g here, f here, i here. Now I have chosen the first row. I did a, I did b, I do c. This tells me that c is supposed to be positive, so I just add c. Again, we're changing colors here. Let's see if pink might be good. When I'm up to my part where c is highlighted, I'm getting rid of this column and this row. What I'm left with is this right here, so I have d, e, g, and h. Now all I need to do is find the determinant of these three two-by-two matrices, which we already know how to do, and the rest of it falls into place. Now, a little note here, keep in mind that I'm not changing the sign of a here to positive. I'm just saying that a will stay whatever it is. I'm keeping it. I'm almost multiplying by positive 1. Here I'm multiplying by negative 1. Here I'm multiplying by positive 1. I'm not saying that if a is negative, it turns positive. All I'm saying is whatever a is, stays. Whatever b is, we negate it. Whatever c is, stays. If c was negative 2, we would still have a negative 2 here. I just wanted to make sure that was clear. Next time, we'll do this concept; we'll bring it to an actual example of a three-by-three. I'll do the example starting with the first row, which is a little bit simpler because it starts with a plus here. I'll do that. Then I'll take the same matrix, and I'll find the determinant starting with the second row, and we'll flush out what these coefficients mean. By the end of the two examples, you should have a stronger understanding of how to figure out a determinant of a three-by-three matrix.
