Upper triangular determinant

let's say I have a matrix where everything below the main diagonal is a zero and I'll start just for the sake of argument let's start with a 2x2 matrix so let's start with a 2x2 matrix so I have the values a B 0 and D so instead of a C I have a 0 there so everything below and the main diagonal is a 0 so what is the determinant of this going to be let's call that matrix a so the determinant of a is going to be equal to ad ad minus B times 0 so that's just 0 so you don't have to write it so it's equal to a times D now let's say I have another matrix let's call it B and let's say it's a three by three matrix so three by three matrix and let's say it's entries are a B see you got a zero here then you let's say you have a D here E then you have another zero here another zero here and you have an F so once again all of the entries below the main diagonal are 0 what's this guy is determinate what we learn several videos ago that you can always pick the row in the column that has the most zeros on it that simplifies your situation so let's find the determinant along this column right here so the determinant of B the determinant of V is going to be equal to a times the sub matrix if you were to if you were to ignore a row and column so a time's the determinant of D e0f and then minus 0 minus 0 times its sub-matrix so you could cancel out or times the determinant of its sub-matrix that row and that column you get b c 0 f vc 0 f and then you have plus 0 plus 0 times you get rid of that row that column you get b c d e b c d e now obviously these two guys are going to be 0 I don't care what these 2 by 2 matrices what their determinants end up evaluating to so these are both going to be equal 0 because we're multiplying by 0 I left with a time's the determinant of this and the determinant of this is pretty straight forward we're going to have it's going to be equal to a times determinant of this is DF minus 0 times Z so it's just going to be DF so it's going to be D so the determinant of B is a DF notice the determinant of a was just a and D now you might see a pattern in both cases we had zeros below the main diagonal right this was the main diagonal right here and when we took the determinant of the matrix the determinant just ended up being it just ended up being the product of the entries along the main diagonal and if you think that that's a general trend that always applies you are correct we can do it in the general case let's do the general case so let's say we have some matrix a and it is equal to it is equal to a11 and then you have a 2 2 you're going to have a 0 right there and then you just keep going all the way down to a and N and this row everything is going to be a 0 everything's going to be 0 except for that last column this is all a 0 right here so everything below everything below the main diagonal is a 0 just like this one but we're doing it in the general n by n case and everything up here is well it doesn't have to be 0 this is a 1 2 all the way to a 1 and this is a 2n keep going down so everything and the main diagonal are above isn't necessarily equal to 0 so if you wanted to find the determinant of a we can do the same thing we did here we could go down go down that first row right there so the determinant the determinant of our matrix a is equal to this guy a11 times the determinant of its sub-matrix so that's going to be a 2 2 goes all the way to a 2 N and then a 3 3 all the way to a and n all the way to a n N and then everything down here is these are all Zero's these are all zero so once again we have another situation where all of the entries below the main diagonal are zero so what's the determinant of this guy right here and what you might say hey what about the rest of that row well the rest of the row is just a bunch of zeros just like we had here it's zero times the determinant of its sub-matrix and then I've you minus and a plus zero times the determinant of its sub-matrix so on and so forth so we just have to pay attention to this term right there now the same argument we can do here to find this determinant we can just go down that row so determinant of this is just going to be equal to let's write out let's not forget our a11 out there a 1 1 and the determinant of this is going to be a 2 2 a 2 2 times the determinant of its sub-matrix get rid of its row and it's column and you're just left with a 3 3 all the way down to a and n everything up here is nonzero this is a 3 n and then everything below the diagonal once again this is just a bunch of zeros everything down here is a bunch of zeros another another what we call upper triangular matrix let me write that down this whole class where you have zeros below the main diagonal these are called upper upper triangular matrices triangular matrices matrices just like that now we keep doing the process over and over again if you just keep following this pattern over again now you're going to have the determinant of this is a 3 3 times its sub-matrix and every time the sub matrix is getting smaller and smaller you will eventually you will eventually get to a 1 1 times a 2 2 times all the way to a n minus 2 n minus 2 times a 2 by 2 matrix over here times a 2 by 2 matrix here that's just going to be a n minus 1 and minus 1 a n sub n this is going to be a sub n minus 1 n and then you're going to have a 0 right here so it's just the bottom right-hand corner of our original matrix is what you're going to be left with and what is the determinant of this well it's just the product of these two things it's just this guy times this guy - this guy times that guy but that's just zero so the determinant of a the determinant of a ends up becoming a 1 1 times a 2 2 all the way to a and n or the product of all of the entries of the domain diagonal which is a super important takeaway because it really simplifies finding the determinant so what would otherwise be really hard matrices to find the determinants of you could imagine if this was you know 100 by 100 matrix now we can just multiply the diagonal so just to make sure that things are clear let me do an example let's say we find the determinant of 7 3 4 - yes so we have zeros here so it's a minus 2 1 and a 3 with a 0 here sorry we don't want zeros there we don't need to have zeros there 6 7 we actually could have zeros there but we don't need to have zeros there the 0 there you have zeroes there just like that so it's upper triangular matrix if you want to evaluate this determinant you just multiply these entries right here so the determinant is equal to 7 times minus 2 times 1 times 3 so it's 7 times minus 6 which is equal to minus 42 and it's that easy