Determinants along other rows/cols

in the last video we evaluated this 4x4 determinant and we found out that it was equal to seven and the way we did it is we went down this first row we use the definition I gave you in the last video where you use this first row I could even write it here we said this is equal to 1 times the determinant of 0 to 0 1 2 3 3 0 0 minus 2 times the determinant you cross these guys cross that row and that column out 1 0 2 1 0 2 2 2 0 0 3 0 and then we went to the plus the 3 times its sub-matrix I don't have to figure that out if you cross out that row that column and then minus 4 we just keep switching the signs times the determinant of its sub-matrix so this one has a bunch of terms and this one's going to have a bunch of terms cross that row and that column out you get 1 0 2 0 1 2 2 3 0 I'll just write here 1 0 2 0 1 2 2 3 0 and that is a completely legitimate way to figure out a determined and that was our definition for how to find terminate but I want to show you in this video is that there's more than one way to solve for determinate what I'm going to show you this way is the same thing that we did down this first row we can actually do down any row or any column of this determinant and of this matrix and the reason why that's useful is because we can pick rows or columns that have a unusually large number of zeros because that tends to simplify our computation so the first thing that you have to do before you embark on picking an arbitrary row or column let's say for example we want to pick this let's do one row in one column in this example so let's say we want to go down that row instead because we like the fact that it has a lot of zeros there the first thing you have to do is remember the pattern remember you switch signs on the coefficients and you don't just switch signs as you go down a row you also switch signs as you go down a column so the general pattern for 4 by 4 will look like this it'll be plus minus plus minus minus plus minus plus and then you get what a plus minus minus plus plus minus minus plus really this checkerboard pattern if you wanted to figure out the sign for any IJ so let's say you wanted to figure out the sign for you know this is 2 2 so if you want to find the sign let me write it this way let's say we have a function let me define a function and I think the checkerboard pattern is pretty clear to you but I'll just write it down so let's say I wanted to figure out the sign the sign not the not the trig identity not the trig ratio I want to figure out the sine of any entry where you give me an i and a J what you can do is you just take negative 1 to the I plus J power so if you wanted to figure out the sign for the let's say you are in you are in row 4 column 2 row 4 column 2 so what will be you do 4 plus 2 1 negative 1 to the 6 power is equal to 1 so that's going to be a positive positive 1 now if you take this guy let's say you want to take let's say you want to take this guy so this is I is equal to 2 J is equal to J is equal to 3 you're in the second row third column 2 plus 3 is 5 minus 1 to the fifth power is minus 1 and you have a minus there so that's another way to think about it but the checkerboard pattern is pretty straightforward so now that you have the checkerboard pattern in your mind let's go down this row let's go down this row so we start with the 2 but notice that we have to multiply it by minus because you go plus minus plus minus so you have a minus 2 times the determinant of its sub-matrix so you cross out this row in that column and you're left with this matrix up here so it's 2 3 4 0 2 0 1 2 3 and then you go you this was a minus so then you have a plus so then you have a plus 3 times get rid of this guy's column and row and you have 1 1 0 1 1 0 that's that right there 3 2 2 3 2 2 and you have 4 0 3 or 0 3 and then would have a - a zero times its sub-matrix plus a zero but we can ignore those because zero times anything is a zero so already we've simplified our determinant a good bit so let's see if we can evaluate this and get the same number because only then will it be reasonably satisfying so what's the determinant of this guy well we can do it the exact same principle we can go down any row or column that seems to be especially simple so let's go down let's go down that row because that row seems especially simple so this is going to be minus 2 that's this minus 2 right here times the determinant of this guy so the determinant of this guy we just have to go and say okay we have a plus and we have a minus and then we have a plus so it's going to be minus 0 times its sub determinant I guess we call it we get rid of that row and or that column in that row would be so it'd be minus 0 times anything this is going to be 0 plus 2 so plus 2 times the determinant get rid of its get rid of its row and it's column so it's 2 4 1 3 2 4 1 3 and then you would have a and then you would have a minus 0 times this thing but who cares what that is because you have a 0 times it so that just simplified to that which is nice let me get write it like that and then you have plus 3 plus 3 times this thing right here let's pick a nice we don't wanna do the first row we have no nonzero terms here let's at least do let me do this row right here for a little bit of variety we've none of the columns actually seem that interesting they all have it at most one 0 so if we do that one right here this is a plus minus plus minus plus so we'll have a plus 0 times 3 4 to 0 we can ignore that minus 2 times the determinant get rid of this column that row so 1 I have to be very careful I put this - there but there wasn't a minus 1 there so let me write it like let me make sure I want to make sure I don't make any careless errors right here this is a plus 1 I was just I just do a minus 1 there to show you how things which signs so this is going to be a this is going to be a 3 times so we're going to go down this is this three right here I lost my bearings with that - there but we were trying to find the determinant of this so it's a zero times this matrix we could ignore that - two times its sub-matrix which is that and that so it's one one four zero and then you have a plus three times its sub-matrix one three one two one three one two just like that and see if we can simplify this 2 times 3 is 6 minus 1 times 4 so this becomes 6 minus 4 so that's 2 so this whole thing simplifies to 2 times 2 which is 4 4 times minus 2 the whole thing simplifies to minus 8 now this guy right here we have 1 times 0 - 0 minus 1 times 4 is minus 4 times minus 2 this whole thing becomes a positive 8 and you have 1 times 2 which is 2 minus 1 times 3 which is minus 3 so you had a minus 1 so this becomes a - I mean you get 2 minus 3 minus 1 times 3 so this becomes a minus 3 so you have an 8 minus a 3 so this becomes a minus 5 and then you have a 3 you have a 3 times you have a 3 times a minus 5 right there so 3 times minus 5 is going to be equal to minus 15 let me make sure I got oh I made a silly mistake if you have an 8 minus a 3 this is 8 this is - this is going to be 5 very easy your brain starts to get fried if you do this long enough and then you have a 3 times a 5 you get a 15 you get a 15 and then you have a 15 this term Plus this term is a 15 - and 8 which is equal to 7 which lucky for us I barely evaded making a careless mistake there we got the right answer but this is a much simpler computation than we did in the last video and it was much simpler because we picked the row that happened to have a lot of zeros on it so we only had two of these terms instead of 4 terms like we had in the last video and you could do the same you could pick columns that have a lot of zeros you just have to make sure that you always use this checkerboard pattern