All right. So we just learned the hypothetical way to find the determinant of a three x three, but it might have been confusing and you kind of have no idea what's going on or maybe you completely understand it. Either way, examples solidify foundational things, right? So let's do an example here and again, I'm going to do this example twice, once with the first row, and then again with the 2nd row, just to show you kind of how the coefficients change. And also in general how to approach this from, two different sides. So here we want to pick the first row, right? That's our foundation here so what we do is we say a and again, let's let's put up that little plus minus plus minus plus minus. And then here it has to be a plus minus plus just for a little bit of reference here. So what we're going to do is and I won't have room to do it here. So come down here, we want the first and we know it's positive. So we're going to multiply by Plus one or we're just going to keep it the same essentially. So we're going to keep it the same because it is positive. We have a plus one already so one times the determinant of when we get rid of this one column and one row we're left with this and just like for the foundational last lesson, I'll colour, I'll try to colour code these. So for one we're left with this, right? We're getting rid of this column, we're getting rid of this row we're left with this little two by two. We want the determinant of that -2 and zero okay, now we go along the row, we've already done the one right here, right? This one is this one now we keep going across the road, we have five. So we put the five here, we keep in mind that this entry is a negative, so we can have a -1. So we're going to do -5 here okay, It's normally plus five, but we have this negative, that changes it. We have the determinant of whatever is left over is over this five. We're going to get rid of this column. Get rid of this row and we're going to be left with this and this right, we're cancelling this column here. We're cancelling this row, the column and the row associated with the coefficient that we're on right now. So we have 20 and we also have -1 Zero. Okay, now the next color had pink around here. Somewhere here it is, let's talk about the zero. It's going to be a zero so nothing will matter, right? Zero times anything is zero. It'll just go away but let's do it either way. So the zero is in this position which is a plus one, times zero. And then when we are on the zero, this column is gone. This row is gone and we're left with this, which is a 2, 4, zero, negative two. So now all we need to do is actually do it. So we know how to do a 2x2 right? When we find the determinant of a two by two matrix it's a times d minus b times C. So in this case we have one out in front times, let's do the determinant of this, it's four times zero. Zero minus negative one negative two. Okay Now we have -5. The determinant of this is two times 0 minus -1 times zero. Alright, two times zero minus negative one times zero. Okay and here we have plus one times zero is zero times the determinant of this which is two times negative two, 2002. Hopefully it's still showing -4 times zero so in this case I'm just going bring this up here to finalize it. This whole thing, we're going to bring this whole term which is the determinant up here one times four times 00. Let's make sure we keep track of all these negatives minus negative one, times negative two is positive two. And we're going to subtract it so this will be minus two. This will be zero. So zero minus two is negative two times one. So we have negative two minus five. In this case two times 00 minus negative one times 00. This whole term zero negative five times zero will give me zero. So plus or zero right? So let's just zero. Doesn't matter and then plus zero times anything is zero. So we know it's again plus zero. So this is -2 so this matrix we have solved using the first row method that the determinant of this matrix is negative two next time. Because this whole thing is a little bit confusing. We figured it out with this, we're going to do it a little bit faster without all the extra notation, like the plus one minus one plus one. And we're going to use a different row which changes up the coefficient next time we'll be using the same matrix but it'll just solidify the idea. And if this example was a little bit, how do I do it in this situation or what's going on here? We're just going to hammer it home a little bit in the brain to get used to doing this.
