Okay. So, we're revisiting the example we did last time. Except this time, we're going to choose a different row. We'll choose this row. And again, just for reference, let's put the plus, minus, minus, plus, minus, plus, minus. Again, you don't need to memorize this. Just always start with a plus in the top left and anything touching it vertically or horizontally should be a minus. And then from there on, no symbol, no plus sign should be touching another plus sign horizontally or vertically. And you can just recreate this for any n by n matrix that you need. You don't need to memorize this. [COUGH], okay. This will just be a little thing on the side for reference here. So, we're choosing the second row. So, we're going to do the same concept we did last time. Except, notice here that we're on the second row which starts with the negative. Okay. So, we're going to be negating the coefficient sign of whatever is here. And then also in the third column in the first row. And if we had more columns, we would just continue this trend on. So, second row. So, in general we have a 2. We start with the 2. But if we go to our little reference matrix we know it's negative, there's -2. And we need the determinant of the 2 by 2 matrix that results when I get rid of the column and row that has this 2 in it. All right, this coefficient whatever I used. So, I use this 2. So, I'm getting rid of the first column. I'm getting rid of the second row. And what's left is this 5,0 and this -2,0, right? I'm getting rid of this column, I'm getting rid of this row. So, what's left goes in here? So, I have a 5, a 0, a -2,0. [COUGH] now, let's use a different color. Now, when I go to the, I'm always staying in the same row. So, I'm on the second row. Going to the next column I have this 4. But this 4 has a plus, which again doesn't mean I'm changing it to a positive or negative, it just means I'm leaving the sign alone. I'm keeping it the same, I'm multiplying by positive 1. However you want to think about it. So, I have a positive 4, it's staying the same. So, +4 times, now this 4 is in the second row, second column, If I get rid of the second row and second column, I'm left with this guy, this, this, and this, right? Getting rid of the column and row associated with this 4. So, I have a 1, a 0 write a 1, a 0 to making a little 2 by 2 matrix for me. 0,0. Good. We're almost there. I've not used pink yet, let's do that. So, now I'm up to this 1. So, this will really kind of show you exactly what's happening here. This negative 1, normally I'd have a negative one here. But my reference chart says that I'm supposed to multiply the coefficient by negative one, right? Or swap the sign on my number here because it's negative. So, my -1 here, turns into a +1. Now, when I get rid of the column which is this column and the middle row. I get rid of the column and row associated with this value, I'm left with this and this. Which again gives me a neat little 2 by 2 matrix, 1, 5, 0, -2. So, again the 2, I changed the sign. The 4 I left the sign. The -1, I changed the sign. Hopefully we're understanding. Because the second row kind of changed everything we needed to choose a different reference row. And so, this is what we ended up with. So, we have -2. The determinant of this is, 5 times 0=0 minus 0 times 9 = 0. So, we have 5 times 0=0. I can write it out- 0 times -2. 0 times -2. So, make sure we keep that negative in mind here. This will just go away because 0-0, but we'll get to that. +4, (1 time 0)=0. So, 1 times 0 minus 0 times 0. That will also go to 0. +1 (1 times -2) is -2. -(5 times 0). So, -2 times now 5 times 0 again is 0 minus 0 times anything is 0. So, this whole thing is 0. So, this whole thing will go away. So, we'll have 0 or -0. However you like to do things. +4 (1 times 0) minus 0=0. This whole thing will go away to 0 as well. So, this whole thing will go away again. 0 + 1( -2 minus 5(0)=0. So, be a minus here. So, we'll get -2. Same determinant, we got the first time when we did it with the first row and if we did it with the third row correctly, we'd also get -2. So, again the same matrix, just how do we deal with this little reference chart of positive negative? How do we translate that into neatly setting up the problem? And then doing the smaller determines. Now, if you had a 4 by 4 matrix, you would have a larger reference chart, right? But we already know how to create that. Or 5 by 5, we'd even be able to create that. It's just alternating plus minuses. We had a larger 4 by 4 matrix, again it would be whatever row we picked, we'd maybe if we pick the first row would be this positive. And then we'd have whatever's left when we got rid of that row and column, which is a 3 by 3 matrix. So, in a 4 by 4 matrix, if you're trying to do the determinant, you're going to have this complicated plus minus thing to keep track of plus. You'll also have to keep track of all these smaller 3 by 3 matrices, right? And you'll have to find the determinant of those. Which as we can see from the simple one determinant of a 3 by 3 matrix, it becomes a lot of work. So, 4 by 4, 5 by 5, they start to become overbearing amounts of almost tedious work. So, 2 by 2 is a great foundation for determinants. Knowing how to use that to find a quick inverse of a 2 x 2 matrix, is very useful at times. And being able to solve a 3x3, is a realistic task in terms of math, linear algebra, and data science. Going beyond that into a 4 by 4 or more might be a little bit onto the side of it's a little tedious. Let's start to use computers. But I think a 3 by 3 is still realistic. So, being able to do this to do this comfortably excuse me, is definitely a good skill to have.
