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Current time:0:00Total duration:14:27

Determining invertible matrices

Video transcript

perhaps even more interesting than finding the inverse of a matrix is trying to determine when an inverse of a matrix doesn't exist or when a when it's undefined and a matrix a square matrix for for which there is no inverse of which an inverse is undefined is called a singular matrix let's think about what a singular matrix will look like and that how that applies to the different problems that we've addressed using matrices so if I add the I'll do a 2x2 because that's just a simpler example but it carries over into really any size square matrix so let's take our 2x2 matrix and the elements are a b c and d what's the inverse of that matrix this hopefully it's a bit of second nature to you now it's 1 over the determinant of a 1 over the determinant of a time's the adjoint of a and in this case you just switch these two terms so you have a D and an A and you make these two terms negative so you have minus C and minus B so my question to you is what will make this entire expression undefined well it doesn't matter what numbers I have if you have if I have numbers here that make a it define then I can obviously swap over make them negative and it won't change this part of the expression but what would create a problem is if we attempted to divide by zero here if the determinant of the matrix a were undefined so a inverse a inverse is undefined is undefined if if and only if and and you know in math and they sometimes write it you know if with two F's if if and only if the determinant of a is equal to zero so another way to view that is if a determinant of any matrix is equal to zero then that matrix is a singular matrix and it has no inverse or the inverse is undefined so let's think about in in in the conceptual terms at least the two problems that we've looked at what a what does zero determinant means and see if we can get a little bit of intuition for why there is no inverse so what is it zero determined well in this case what the determinant of this 2x2 what the determinant of a is equal to what it's equal to ad minus BC equal to a D minus BC so this matrix is singular or it does has no inverse if this expression is equal to 0 so let me write that over here so if a d is equal to BC or we can just manipulate things and we could say if a over if a over B is equal to C over D where I just divide both sides by B and divided both sides by D so if the ratio of A to B is the same as the ratio as C to D then this will have no inverse or another way we could write this expression if a over C if I divide both sides by C and divide both sides by D is equal to B over D so another way that this would be singular is if and it's actually the same way if this is true then this is true these are the same just a little bit of algebraic manipulation but if the ratio of A to C is equal to the ratio of B to D and you can think about why that's the same thing as the ratio of A to B being the same thing as the ratio of C to D anyway I don't want to confuse you but let's think about how that how that translates into some of the problems that we looked at so let's say that we we wanted to look at the problem let's say that we had this matrix representing the problem the linear equation problem well actually this could be either one so where we have a b c d x x y is equal to i don't know two two other numbers that we haven't used yet a and f so if we if we have this matrix equation representing the linear equation problem then the linear equation problem would be translated a times X a times X plus B times y plus B times y is equal to e + C times X plus D times y is equal to f and we would want to see where these two intersect that would be the solution the vector solution to this equation and so it just just to get a visual understanding of what these two lines look I let's put it into kind of the the slope y intercept form so this would become what in this case y is equal to what Y is equal to minus a over B minus a over B X plus a over B right I'm just giving some steps when you subtract a X from both sides and then divide both sides by B and you get that and then this equation if you put it in the same form just solve for y you get Y is equal to what minus C over D minus C over D X minus C over DX plus F over Y so what do you what let's think about this let me I should probably change colors because it looks to let's think about these two equations what these two equations would look like if this holds right when we said if this holes in them then we have no determinants and this becomes a singular matrix and it has no inverse and it's and since it has no inverse you can't solve this equation by multiplying both sides by the universe because Z inverse doesn't exist so let's think about this if this is true we have no determinants but what does that mean intuitively in terms of these equations well if a over B is equal to C over D if a over B is equal to C over D these two lines will have the same slope they'll have the same slope so if these two expressions are different than what what do we know about them if two lines that have the same slope and different y-intercepts they're parallel to each other and they will never ever intersect so we could let me draw that just so you get the so fats you know so line this top line might look what they both have let's say these are pie they don't have to be positive numbers but just has a negative I'll draw it as a negative slope so that's the first line and it's y-intercept will be its y-intercept will be e over B Y over B that's this line right here and then the second line now let me do another color would look something like I don't know it's going to be above or below that line but it's going to be parallel it looks something like this it looks something like that and that lines y-intercept so that's this line that lines y-intercept is going to be f over Y so if Y over B and F over wire are different terms but these both lines have the same equations they're going to be parallel and they'll never intersect so they're actually would be no solution if someone told you well you know just a traditional way that you've done it either through substitution or through adding subtracting the linear equations you wouldn't be able to find a solution where these two intersect if a over B is equal to C over D so one way to view the singular matrix is that you have parallel lines then you might say hey Sal but these two lines would intersect if I over B equals F over Y if this and this were the same then these would actually be the identical lines and not only would they intersect they would intersect in an infinite number of places but still you would have no unique solution you would you would have no one solution to this equation it would be true at all values of x and y so you can kind of view it when you apply the matrices to this problem the matrix is singular if the two lines that are being represented are either parallel or they are the exact same line the parallel and not intersecting at all or they're the exact same line and they intersect in an infinite in an infinite number of points and so it kind of makes sense that the a inverse wasn't defined so let's think about this in the context of the of the linear combinations of vectors that's not what I wanted to use to erase it so when we think of it when we think of this problem in terms of linear combination of vectors we can think of it like this this is the same thing as the vector a C times X plus the vector B D times y is equal to the vector e F so let's think about it a little bit we're saying is there some combination of the vector AC and the vector BD that equals the vector EF but we just said that if we have no if we have no inverse here we know that because the the determinant is 0 and if the determinant is 0 then we know in this situation that a over C must equal B over D so a over C is equal to B over D so what does that tell us well it essentially tells us that well let me let me draw it and maybe numbers would be more helpful here but I think you'll get the intuition I'll just draw the first quadrant I'll just assume both vectors are in the first quadrant let me draw so the vector AC let's say that this is a let me do it in a different color so I'm going to draw the vector AC so this is a and this is C then the vector AC looks something like that let me draw it I want to make this neat the vector AC is like that and then we have the arrow and what would the vector BD look like what is the vector BD look like well the vector BD well unless you know I could draw it arbitrarily someplace but we're assuming that there's no there's no derivative sorry there's no determinants have I've been saying to remember the whole time I hope not well we're assuming there's no determinant to this to this matrix so there's no determinant we know that a over C a over C is equal to B over D or another way you could view it is that C over a is equal to D over B but what that tells you is that both of these vectors kind of have the same slope so if they both start at point zero they're going to go in the same direction they might have a different magnitude but they're going to go in the same direction so if this is if this is point B and this is point D vector BD is going to be here you might just want to think if that's not obvious do you think a little bit about why these two vectors if this is true are going to point in the same direction so that vector is going to essentially overlap is going to have the same direction as this vector but it's just going to have a different magnitude it might have the same magnitude so my question to you is vector EF we don't know where vector EF is well let's say I don't let me just pick some arbitrary point let's say that this is e and this is f so this is vector EF up there I'm going to in a different color vector EF let's say it's there so my question to you is if these two vectors are in the same direction maybe a different magnitude is there any way that you can add or subtract combinations of these two vectors to get to this vector well no you can scale these vectors and add them and all you're going to do is kind of move along this line you can you can get to any other vector that is a multiple of one of these vectors but you can't get because these are the exact same direction you can't get to any vector that's in a different direction so if this if this vector is in a different direction there's no solution here if this vector was if it just happened to be if it just happened to be in the same direction as this then there would be a solution where you could you know just just scale those actually there would be there would be an infinite number of solutions in terms of x and y and then but if if the vector is slightly different in terms of its direction then there is no solution there's no combination of this of this vector this vector that can add you up to this one and it's something for you to think about a little bit it might be obvious to you but the other way to think about it is when you're trying to take sums of vectors you know any any other vector in order to move it in that direction you have to have a little bit of one direction and a little bit of another direction to get to any other vector and if both of your kind of ingredient vectors are the same direction there's no way to get to a different one anyway I'm probably just being circular in what I'm explaining but that hopefully gives you a little bit of an intuition of well one you now know what a singular matrix is you know when it's when you can when you cannot find its inverse you know that when when the determinant is zero you won't find an inverse and hopefully and this was the whole point of this video you have an intuition of why that is because if you're looking at the vector problem there's no way that you can find that you'll either that there's no solution to finding a combination of the vectors that gets you to that vector or there an infinite number and the same thing is true on finding the intersection of two lines they're either parallel or they're the same line if the determinant is zero anyway I will see you in the next video