43 - Properties of determinants

what I want to do now is collect a bunch of rules for calculating determinants I'm calling it properties of determinants some are going to be completely trivial like the first one I'm gonna write others may require even to believe them require some thinking but we're not gonna do a lot of proofs here we're just really listing them it's like a big shopping list of rules okay so here's the first property and this you should agree is obvious if one of the rows or columns is all zeros then what is the determinant right then the determinant is zero because choose that row or that column which is all zeros and use that so you're going to have zero times a minor minus 0 times a minor plus 0 times a minor minus 0 times a my obviously it's just gonna be all zeros so this is a trivial statement once you know that you can develop the determinant using any row or any column good so I'm even going to skip giving you an example this is really straightforward ok property number two the determinant of triangular matrix triangular matrix is the product of the main diagonal entries I'm gonna convince you that this is true using an example okay so late let's take four by four triangular matrix here's a triangular matrix a triangular matrix can either be upper triangular where all the entries above the main diagonal are are non zero or lower triangular where all the entries below the main diagonal are nonzero so this is going to be a lower triangular matrix 2 negative 3 0 0 4 negative 1 3 0 and 2 negative 5 1 2 how do I find the determinant how do I calculate it I have to choose a row and choose a column let's choose the first row all of these contribute nothing so the determinant is just 1 times the corresponding minor the corresponding minor is cross out this row in this column you get left with this negative 3 0 0 a negative 1 3 0 and negative 5 1 2 good again I have to choose a row or a column I'm not going to be at all creative I'm gonna choose this what do I get I get these two contribute nothing I just get negative three let me write this as 1 times negative 3 times this minor 3 0 1 2 that's it and this is just 3 times 2 minus 0 times 1 so I get 1 times negative 3 times 3 times 2 you can calculate this number it happens to be what 6 times negative 3 negative 18 but that's not what I care about what I care about it's that it's just the product of the main diagonal entries good ok and this is this the proof of this statement would be precisely writing this abstract the a 1 1 a 1 2 A 1 3 a 1 for up to a 1 F a 2 1 a 2 2 and doing this exact thing you can see that it's tedious but there's nothing to it there's no no further theorem to use or anything bright to say it's just what we did in the example to spell it out abstractly in general okay next so this was property number 2 property number 3 if we exchange two rows or 2 columns in in doing row operations exchanging two rows was a legitimate row operation meaning that for example the two matrices had the same null space represented the same set of solutions okay for a system when you exchange two rows in a matrix it doesn't preserve the determinant the determinant changes but not too significantly only by a sign okay so if we exchange two rows the sign only the sign of the determinant changes so if the determinant was negative five you change two rows the determinant is gonna be five okay right if you do it twice you did it's like doing nothing okay so example let's take this matrix I'm taking purposely something very easy okay so the determinant here is one I can see it right away because it's a triangular matrix so it's the product of the diagonal entries let's change two rows let's change for example what did I change the first row in the second row so 0 1 0 1 0 0 & 0 0 & 1 ok I switched two rows what's the determinant now well I have to choose how to evaluate it let's evaluate it for example using column number two any choice would be good here because there are plenty of zeros all over the place so what do we get it's 1 times 1 0 0 1 but it's a even column so we start with the - good so that the sign flipped when we exchanged locations of two rows number four if a has two equal rows or columns then the determinant of a is necessarily zero why is that true why is that true suppose you have I'm justifying it I'm not gonna write it but here's the idea suppose there are two equal rows okay the determinant is something let's call that something s for something the determinant is something when I exchange the two rows the determinant is going to become minus whatever it was because the change the sign changed by three so like like this the determinant was s now the determinant is minus s but these were equal rows it's the same matrix so for the same matrix the determinant is s and minus s simultaneously that can only happen if s is 0 do you agree ok so idea idea of proof exchange the two rows the two equal rows by 3 by 3 you would get that a equals minus a right but this equality a number equals it's negative if and only if that number happened to be 0 this is the idea of the proof it's it's everything okay you need to do to in fact disapprove it's really the proof no there isn't if it has two equals Rose doesn't mean there's a row of zeros all I'm saying is that two rows are the same okay okay property number five a common factor of a row or a column can be taken out of the determinant what do I mean by this let's do an example let's look at the matrix 1 2 3 8 10 12 7 8 9 well be careful with the word divide do you see that row number 2 has a common factor of 2 okay this is 2 times 4 this is 2 times 5 and this is 2 times 6 ok so this equals 2 which I pulled out as a common factor from the second row times the determinant of 1 2 3 4 5 6 7 8 9 okay and we know that this is 0 therefore this is 2 times 0 which is 0 this is because we we calculated this determinant earlier and we saw that it was 0 okay so that's what I mean by a common factor of a row or a column can be taken out or pulled out of the determinant ok again this is different than row operations ok in row operations multiplying a row by a scalar didn't change the the system for example here it does a scaler shows up ok here the determinant was something the determinant of this is 2 times 2 something ok word of caution we had advice now we have word of caution suppose we have alpha a alpha C B and D the determinant of this thing you can pull out the Alpha right so it's alpha ABCD I pulled out a common factor out of the first stroke right what is the word of caution what what is the word of caution that the scalar here that was pulled out of the determinant didn't multiply all the entries like when we multiply a matrix by a scalar it only multiplied one row or one column okay so it's not true this is true so in general in general if we take a matrix a and multiply it by a scalar alpha multiplying the matrix by a scalar is multiplying every single entry by that alpha this is not the same or let's write what it is the same it is the same as you'd have this alpha in every row and in every column so you would pull it out how many times n times one from each row so this is the same as alpha to the power n times the determinant of a and it is not the same and alpha a is not the same as alpha a that's the point okay so scalars do not just hop out of determinants they hop out with with the power and if they're on every entry okay and they hop out once if they're only in one column okay so this is the word of caution this you can call a property okay so this is in fact a property that follows from number five good clear okay by the way why is this true this is pretty easy to convince yourself that it's true suppose we were to evaluate using this row we would get precisely two times the miner times for 2 times the miner times 5/2 times the miner times 6 we would get precisely 2 times the determinant of this do you see that do you see why this is true ok every month if we were to evaluate using exactly this row it's completely obvious that this is what we would get okay and that's the way a general proof would go okay clear okay where are we number 6 property number 6 if one row or a column is a multiple of another row or column butt-butt-butt respectively so if one row is a multiple of another row or one column is a multiply of another column then the determinant is zero oh this is another way of denoting the determinant I didn't mention it okay I used only the the vertical bar notation this is an equivalent way of denoting the determinant you'll see this in many books or people use it and okay good happens by mistake but it's good that it happened okay so why is why is this true so if one Row is a multiple of another row right take out the scalar multiple you'd be left with to eat rose and when there are two equal rows we know that the determinant is zero do you agree okay so idea take out the scaler be left with equal rows okay this is very unfortunate of exchanging grows in property number five we had the row operation of multiplying by a scalar and now what is the next one add a scalar multiple of one row to another okay so if we add a scalar multiple of a row or column to another row or column then the determinant doesn't change so this is in fact an operation which the determinant doesn't sense it's invariant okay so let's see an example I need a new board in fact I need a new marker this one is pretty much dead let's go here so here's an example 1 2 3 4 5 6 7 8 9 so let's add to Row 1 Row 2 and Row 3 so what I'm doing is r1 I'm replacing it by r1 plus r2 plus r3 do you agree that that's adding scalar multiples of rows one to another okay so what do I get I get 12 don't tell me if you if my calculations are correct twelve fifteen eighteen four five six and seven eight nine do you agree okay and now observe that Row 1 is 3 times Row 2 therefore the determinant is 0 right if one road that was property what property 6 if one Row is a multiple of another row than the determinant is 0 right because we can pull the three out and we have two equal rows good ok so this is just an example of this of this property of this fact I want to show you another example here's another example of using these facts that we accumulated to do calculations and and the point is that all these little things are are our little tricks little things you can do to simplify calculations you don't need to calculate all these enormous lis many minors every time if you can observe any one of these facts or dude little operations and just just pay attention to what you're doing right if you're pulling out a scaler you have to remember that the scaler is there it doesn't it doesn't it does change the determinant okay if you switch two rows that's fine as long as you remember to put a minus sign okay so here's an example most of our examples so far were over the field of real numbers let's do a complex example okay to maintain the flavor so here 1 plus I 1 1 1 let me erase this it's in the way let's erase everything I'm gonna need some space for this so it's just right up here so it's gonna be a 4 by 4 example 1 plus I 1 1 1 1 1 minus I 1 1 1 1 1 plus I 1 and 1 1 1 1 minus I so there are no zeros here so it wouldn't be that wise to just start evaluating because you'd have Plitt for 3x3 minors and it would be a lot of calculations ok but what we could do what we could do is take r1 and replace it by r1 minus r2 plus r3 minus r4 okay we know that doing these things doesn't change the determinant this is a adding multiples of other rows doesn't change the determinant so this is an equal sign and what do we get so 1 plus I minus 1 is I plus 1 minus 1 so we just get I do you agree and now I'm not going to do all the calculations you can check me out so you get is I I uh yeah yeah they take sorry 1 1 minus I 1 1 1 1 1 plus I 1 and 1 1 1 1 minus I now I can pull out an eye right there's a common factor in this row I can pull it out remember there's an eye here and I'm left with 1 1 1 1 1 1 minus I 1 1 1 1 1 plus I it looks like this is gonna take take a while right we didn't what what happened only this one changed this one plus I changed to a 1 it looks like nothing much happened so far but in fact we're very close to wrapping this up now subtract the first drove from all the rows ok so our two is gonna be R 2 minus R 1 again these are operations that don't change the determinant right and we get this eye that already lived outside here 1 1 1 1 and then we get plenty of zeros 0 minus I 0 0 0 0 i 0 0 0 0 negative I now it's a triangular matrix the determinant is the product of the diagonal entries so I get I that lived out here times 1 times minus I times I times minus I so 1 times minus I is minus I times I is my I squared which is minus minus 1 which is 1 times minus I so it's I times minus I and this is again just 1 and we're done good ok so here's an example of a calculation that could be tedious if you were to calculate plenty of 3 by 3 determinants here 4 of them all with complex numbers and then these would start multiplying and the expressions would get kind of huh and if you try fooling around with it and try things it's really a trial and error sort of approach you can often save yourselves a lot of work ok so I want to mention two two things then we'll have to end and we'll continue with determinants in a separate lesson we still have some stuff to say about them but just two more remarks so remarks one if we do row operations the determinant changes that's a common mistake to do row operations and claim that the determinant remains the same that's not true ok it doesn't change greatly but you must pay attention to plus/minus --is that occur when you interchange two rows or two columns and n scalars which come out of every single row or column in which they are a common factor okay so it's not true that row operations or column operations do not change the determinant it does change by these so that's the first comment that I wanted to make but if a and B are row equivalent the first comment was they don't have the same determinant they don't have necessarily the same determinant but the determinant of a is zero if and only if the determinant of B is zero and this is going to be a very useful fact later so the determinant itself is not an invariant but being zero is why because if the determinant of a is zero okay all these operations plus or minus Singh zero would not change it would keep it a zero and pulling out scalars would not change it because you'd get five times zero times 7 times zero times whatever so these row operations which is what you do to transfer from A to B using row operations one of them doesn't change the determinant at all that's adding scalar multiples of row and the other to only pull out scalars or changed the sign but doing that to zero does not affect it okay so pulling out scalars multiplicatively does not affect the zero is this clear okay so this is going to be a very useful fact I would actually call it a theorem okay but I wrote it as just a remark to row equivalent matrices do not have the same determinant but if one is zero then so is the other okay good okay so we'll stop at this point and where we're heading what we want to discuss next there are some more properties that I have to mention for example what is the determinant of a product we didn't say that okay and there are some more properties and in particular we want to look more carefully at the relation of determinants to inverse matrices okay and I'll let let me just give a big spoiler remember that serum with the four equivalent statements so there's going to be a fifth one a is invertible if and only if the determinant is not zero okay and then we're gonna see using determinants in fact a method for solving equations a to a different method and for and and and in fact yeah but let's leave some stuff for for next time okay so we're stopping here

Channel: Technion

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Keywords: Technion, Algebra 1M, Dr. Aviv Censor, International school of engineering

Id: 19qaYFGZ-cA

Length: 31min 32sec (1892 seconds)

Published: Thu Nov 26 2015