hi guys this is a loom channel I'm Vincent Conway and I will be taking you to from for mathematics and today we are going to start with the matrix and transformation stay tuned as I welcome you are not refer transformation now we shall look at triangle ABC triangle ABC with the vertices n with the vertices air as 5 6 FP v 2 and C 2 B 2 2 let's say you are given this triangle and you have been told it is transformed through matrix of a transformation 0 negative 1 and 1 0 if you subtract this triangle to this matrix you will find any in a certain image let's see what kind of transformation is this mattress representing so 0 negative 1 1 z you pre multiply it with a given triangle or any given figure you will get a certain image and it will tell you the type and a kind of matrix of transformation you are using for instant here is our triangle ABC with these vertices being subjected to a transformation of 0 negative 1 1 0 when you pre multiply it with this you get the images us negative 6 negative two negative two and this one is five two two five this is a prime B Prime and C prime so we are find that if matrix a if triangle a PC with these vertices is subjected to a transformation of this one we get the image of that the image of that triangle to be negative 6 5 negative 2 5 and negative 2 2 let's see on a Cartesian plane how does this image and the object of this triangle looks like and uploading the images this their a-b-c hello on plotting the image of the triangle a PC that's a prime B prime C prime we shall get you we shall eat the object being as that one so as you can see we can notice that this object is still the same size as the other image yep but the only thing that has changed is the figure in which the manner it looks here and here and we can see that on pre multiplying it with a translation of 0 negative 1 1 0 it has been changed to this kind of image so let's see what kind of transformation is this so any kind of transformation if you are pre multiplying a matrix by any 2x2 matrix you will find that all of them originate from the origin or a passes through the origin for instance this is our origin where it is 0 0 and if you want to find which kind of matrices that then you have to join the two lines using that is from torque like this one you construct a perpendicular bisector to eat and also for example here B you construct a perpendicular bisector you join me to be and you also construct perpendicular bisector to eat using your way of compasses and ruler and you will find that they're both coincide at the origin by the center and on this point if you want to see which kind of transformation is that you can see here that this one has been up to this one so meaning if you take this point from here to here you can see that this is a perpendicular of that one so meaning this one has been transformed from a PC to another triumph of a prime B prime C Prime through a positive product and go in the other direction that is positive for tatin because it's positive 90 degrees at the and with that we can say that we have find the matrix of a transformation to be this matrix it will be a positive for tatin matrix any object being transformed with this kind of matrix it will undergo a positive or a rotation of positive what a can which is positive 90 degrees from the first quadrant where ii now i can say that if you have given you have been given any kind of matrix or any kind of transformation if you want to know what kind of matrix is that or what kind of transformation is it you take any object and in precisely we always take our ones unit square and we multiply it using this kind of transformation or any kind any given kind of transformation for instant here we have this one and we have seen how it works it will always work to the other transformation you are being subjected to I will be taking you through the second subtopic that is finding a matrix of transformation and here finding a matrix of transformation you will be given an object and also an image and will be asked in which kind of transformation did this object passed in order to give this kind of image for instant have been given triangle a triangle ABC with a matrix or oppositional vectors or positional vectors yeah of 1 4 1 1 and 3 1 as the values of a B and C and here DNA image the image of that triangle to be a B C F prime B prime C prime to be having negative 1 4 negative 1 1 and negative 3 1 yeah as negative 3 1 that's the value of C Prime so you will be told find the matrix of transformation for a triangle ABC that turn from this object to this image first we need to decide the kind of our transformation that this underwent and we can say let our transformation t take the coordinates or the matrix a b c and d this is not matrix of transformation and we know from our first session that if you take any kind of transformation you pre multiply it with a given figure or a given quadratic triangle it will give you a given image for instance here we have been given the object of that triangle and here the last few lattices of the image of that triangle so meaning that if we take our transformation as ABCD we pre multiply it with this given object we shall get our image let's see if we take a pc t we pre multiply it with 1 4 1 1 3 1 as the vertices of a b and c we shall get the vertices of a prime negative 1 4 and that of B prime negative 1 1 and C prime as negative 3 one now we needed to find what's the value of F what is the value of P and what is the value of C and also the value of D and from here we know to get this value here we have to take the row multiplied by the column of this matrix and we shall happy having this for instant a multiplied by 1 we shall get a plus this one which is for P now it will be a plus for P it will give you a for P it will give you what negative 1 and to get the value of this one now it will be this that innominate now we are in with the denominator the downward what is that is C multiplied by 1 plus 4 multiplied by T that is C plus 40 is equal to 4 and again if we want to find the value of P prime B Prime we shall get a multiplied by 1 plus P multiplied by 1 and we shall get negative 1 meaning that our a plus B is equal to negative 1 and again if we need this one it will appeal now this one C multiplied by 1 plus D multiplied by 1 and we get that is C plus P is equal to what 2 1 together can't be done two final one which is a multiplied by 3 plus B multiplied by 1 that is 3 a 3 a plus B is equal to negative 3 the remaining will be C by 3 D by 1 and you get 3 C plus T is equal to positive 1 now we need to find the value of the individual components of a B and C together with D let's see we now collect the like terms here for example we can take a plus for P because this has a P this one has we take them together and resolve them small taneous Li for instant a plus 4 P is equal to negative 1 and we also take either a plus B equals negative 1 or we can also take 3 a plus 4 a plus B equals negative 3 together with this order they add any should work so if we take this one and we we solve it is multi Gnaeus Li we shall find that here at will minute this a we need to subtract now to find the value of now it appear and we need to subtract to eliminate a and on elimination we shall find that if you take 4 P minus P you will get 3 P and negative 1 minus minus 1 is equal to 0 because this is negative 1 minus minus becomes positive and it will be 0 so meaning that our 3 P is equal to 0 the value of P now will be you divide both side now 3 P equals to 0 meaning that be it will be used by both sides by 3 to get the value of P as 0 if the value of P is 0 then if you substitute you take one of the equation 8 this equation or the other you find for the value of a now let's see what's the value of a for instance that is a a plus 4 but B what is our P B equals to 0 74 times 0 equals to negative 1 if you multiply 4 by 0 you will get 0 and a will be equal to negative 1 now we have find our value of a up a being negative 1 and P equals to 0 we can still go ahead and find the value of C and B for instance if we want to prove this one is right we took this and this and here we still have that one so meaning that if we take 3 3 a + b she'll give us negative 3 and here we have negative negative 1 so it is 3 multiplied by negative 1 plus B which is 0 and that one gives you 3 negative 3 so meaning that one that method was right now let's find the value of C and the value of D in instant here we have C + 40 here we have 3 + T equals to 1 and here it is 3 C plus B is equal to 1 now we can take 8 at this and this or this and this or the other way round any to work for instant let's take C plus 40 equals to 4 C plus 40 equals to 4 and also texty + T equals to 1 C + D equals to 1 we can still solve this one simultaneously to find the value of C and T now let's subtract to eliminate C if we eliminate this one it will be 0 plus 40 minus D you will get 3t + 4 minus 1 it is equal to 3 now we have 3 T equals to 3 if we need the value of we needed to divide both sides by 3 to get D equals 2 this one cancels with the other 3 3 by 3 you get 1 so our e is equal to D equals to 1 we find our value of B equal to 1 so if we need the value of C we now take any of pay question we substitute P equals to 1 and it will give us the value of C for instance let's now take C plus D equals to 1 now if we take C plus P equals to 1 and we substitute a place of P with 1 we get that C plus 1 equals to 1 implying that C equals to 1 this one data side this one if we call the other side it will be minus 1 and we get our C equals to 0 now we have find a value of T and we have find the value of C so meaning that if we took these values we substitute it into this equation because we use this and this it should give us 1 for instance C equals to 0 take 0 multiplied by 3 you get 0 plus P our T equals to 1 D equals to 1 p 0 plus 1 is equal to 1 meaning our equation and okay so we have found the value of a equals 2a equals to negative 1 B is equal to 0 C is equal to 0 and P is equal to 1 now we can conclude that our transformation apct it is these values now our T becomes we substitute the values of a b c and b in this matrix and we find it will be equal to negative 1 B 0 C and E 1 this is our transformation and if we needed to know the type and kind of transformation you can use the first procedure I use to identify by drawing a Cartesian plane now this image at the object and the image you will find what kind of transformation this represents that was the first way in general we can say that if you have been given given the object or the vertices of an any kind of object be equal the lateral or a triangle and you have been given the image vertices that has passed through a certain transformation T to get that T of which is equal to ABCD you need to pre multiply it with a given object you pre multiply the object vertices with the image with the transformation to get the image and here if you have been given this image vertices and outlet vertices you can still get the transformation by pre multiplying it and a divided floating on a Cartesian plane or by working it out and I can conclude by saying that is how a transformation or a matrix of any kind of transformation is found our next action we shall pass through successive flux a successive transformation under successive transformation we can say we have seen what transformation is and what it does to our given object now in successive transformation simply means you may be given several kinds of transformation being operated on a given object and this object passes through all these theories of transformation even to become a given image for instance having let's say if our first transformation our first transformation P rotation in positive what a tan our first transformation is a rotation in positive quarter turn and another transformation Y being for example let's say 0 1 1 0 this is another knees are flexion now you'll be given a triangle or any kind of object may it be a rectangle square or a trapezium or a parallelogram me I will take a triangle with vertices the aapc having a matrix of 1 4 1 1 and 3 1 now you have been told or given that find the image formed or the image of the triangle ABC through a successive transformation of X X Y has been given here now being given this one it may be X Y or other kind of maybe 4 3 4 5 it may be even 6 but here we are being given true in order to find the successive transformation of this triangle through these transformations x and y we shall first perform this little fast to be performed why is fastly performed then X precedes a proceeds after Y what do I mean I mean given any kind of transformation in that transformation you are being given is the first one to be performed and the outside it will be the last that means that if you want to transform this one through these two transformations then it means we shall first subject a VC triangle a PC through a transformation transformation why will subject triangle a PC through transformation why and we shall first find the first image that is 0 1 1 0 & 1 4 1 1 3 1 and we get the first image of the triangle a prime B prime C prime p 0 x 1 1 by 4 we get for the other one applies that is 1 1 and 1 x 4 pi 0 you get 1 1 1 1 1 3 now we will find that if triangle ABC is path through the first transformation Y we were taught to transform it through X Y is passed through the transformation Y we shall get the first image 3 this but we have been told to transform it through transformation X Y now our Y our first transformation K was this image now we shall now transform our image through the second second transformation and that means that we shall now take triangle a prime B prime C prime through a transformation of we shall take it through a transformation of now X which is 0 negative 1 1 0 and we shall now get negative one negative one negative three and this is now a prime a 2 prime B 2 Prime and C 2 Prime and this if we need if we need the coordinates of here and now it will be 1 by 4 and 0 by 1 1 4 it will be 4 + 0 4 1 by 1 + 0 by 1 we get 1 1 by 3 + 0 you get 1 and now we find that our final image which is a 2 prime B 2 prime C 2 prime we get our final image being negative 1 negative 1 4 negative 1 1 negative 3 1 and we can say that transform the triangle ABC with this kind of vertices as an account through transformation XY to give an image of a 2 prime P 2 prime C 2 Prime and now we can conclude that this this image is the image of triangle ABC through a successive transformation of X Y in general we can say that given any kind of transformation it can be X Y T or even for X Y TV whereby X has been given as 1 0 0 1 and Y can be given as 0 1 0 0 1 1 0 and T be 0 negative 1 1 0 and T and V T V P 1 3 0 1 this can be just an example of the transformation you may be given and then you are being told find find the image of a certain triangle or a given quadratic after passing through our transformation of X Y T this is a successful transformation it will simply mean you find the transformation you firstly perform the transformation of T on that given object say triangle a PC with the be given vertices you firstly perform T on it then after performing T you perform as you go backward you perform T fast with a given transformation then after getting that image you now again subjected to another transformation which is why with this transformation or coordinate and then after finding that one you take the image you transform it using the last transformation which is X now meaning that when you having in this way you firstly perform the outward or the outside going inward that is you move from your right hand to your left hand then the final image will be given after performing the last transformation which is X same applies to this or any kind of given transformation that is what we call successive transformation performing the first transformation then the other one in order from the right hand side to the left hand side now we in our previous session we already seen in how we can perform our successive matrixes or a successful transformation of a given triangle or a given contractor or any given object and we get their image through that successive transformation and here we now need to see if we have been given different kinds of transformations meaning that one or two or three transformation being given and we needed to transform them or perform them to a triangle or any kind of given figure how my what can each result true now for example being given as x equals to negative 2 0 0 negative 2 and maybe our y equals to 0 1 1 0 now let's say these are our two transformations that is x and y we have seen in our previous session how we can use them to transform a certain given object to a given figure now we need to figure out and find a single matrix that can represent both of them the two of them asked one and transform a given figure or a given object to find the same result as transforming each of them to that object for an instant let's say we have been given a triangle with the vertices a triangle with the vertices a s 1 for B as 1 1 and C as 3 1 now in our previous session we saw how we can transform this triangle using this one for example if we built tall transform triangle ABC through a successive transformation X Y we have already done it in our previous session and now here we shall see how can we transform this triangle using a single matrix X Y as single and give the same image as transforming each of them on this triangle now we shall find a single matrix of these two successive transformations and now having given this X and given Y we can find X Y to be our T where T stands for our single matrix in this case now T will be obtained by taking the matrix of X multiplied by that of Y but in our previous session we said if we are transforming one of them directly to triangle we start from the right-hand side today left you start with this one and then this going the other way but if you want to find the transformation a single transformation you will start this way that is from the left hand side go into the right hand side now meaning that if we want a single transformation for both of them we shall start with negative 2 0 0 negative 2 then 0 1 1 0 if given this one now we we can find T which is a single matrix of transformation for two successive transformation in this case as if you multiply here negative 2 pi 0 plus 0 PI 1 you get 0 and then negative 2 pi 1 by a plus 0 PI 0 you get negative 2 0 PI 0 plus negative 2 pi 1 you get negative 2 & 0 PI 1 plus negative 2 pi 0 you get 0 now we found our T which is a single matrix of transformation of the two successive transformation in this case as this one now we can transform triangle ABC using this one and get the same result as transforming it through fast this one then this one now when you select 0 negative 2 negative 2 0 now to a triangle ABC which can be written with a positional vector of 1 4 1 1 & 3 1 we shall find that the object this is a PC you shall find the image being as 0 PI of Z 1 and then plus negative 2 pi 4 you get negative 8 0 PI 1 2 pi this you get I 0 PI 1 plus negative 2 pi 4 we get the first coordinate be negative 8 and then 0 by 1 plus negative 2 pi 1 we get the second coordinate being negative 2 0 by 3 plus negative 2 by 1 we shall get our third coordinate here being negative negative 2 now technology coordinates we shall find them we shall find them by doing the same by taking negative 2 by 1 plus 0 by 4 and we say this is negative 2 negative 2 by 1 plus 0 plus 1 as 0 by 1 we shall get negative 2 negative 2 by 3 plus 0 by 1 we shall get negative 6 and we have found that this transformation with a single mark a single matrix we have found that it transformed this triangle ABC from this coordinate to an image let's say this is double prime because it has passed through two successive transformation that is x and y we find this way now as your homework or tests I will give you this way now transform this triangle ABC through the first transformation X a Y then you do a stat sweeping transformation X and see if you shall get the same result thank you now we can generally say that given given different kind of matrixes we can still find the a single matrix of those transformation for instance if you have been given X as we can we have seen that X is our X was negative 2 0 0 negative 2 and Y being 0 1 1 0 and a let's say W you have been given as 1 0 0 1 and evenly let's say T being cos 30 or cost 0 cause zeros negative sign zero cost Z sine zero cost zero you will find that if you are being this is none than one zero like the other one this one and this one is Delta semi just confusion but it will take you through the same but the result will be different but if even this one transformation you can still transmit them as required for example you can be told find a single matrix of transformation of XY Y W or even the W Y X you can still find a single matrix of this one by multiplying this matrix you start with the P then the balloon then you perform that of Y then X what do I mean you start with the first one on the second you try you multiplied the first two then the result you multiplied by the third then the result and then with that for what you will have fun is now the single matrix of transformation it doesn't matter how many matrices you have been given any can work together as the same as given this one you start with this one to this you take this you multiplied by this and then you multiply this the result of this one by this W now you will have found a single matrix of transformation now we shall see we are going to pass through our feet subtopic that is the inverse of transformation and here we can see that if given any kind of transformation you can still find the inverse of that transformation in that when you transform that object to a certain image we can find a transformation which will map the same image back to the object for a instant if you have let's say given a triangle ABC with the vertices ABC as for one one for not phone one for me and then 1 1 1 3 3 1 given this one we can transform it through a matrix X as negative 1 0 0 1 and get an image if you transform this one this triangle using the method I have shown you with this transformation - it will give you negative 1 negative 1 but if trade for 1 1 now we can see that this is the image of the triangle L a PC with their with the devadasis a prime B prime C prime giving being given as below now we can say that here it is our matrix of transformation X it has transformed this one but it has transformed triangle ABC 2 image which is this one now we need a matrix which can transform this triangle a prime B prime C Prime back to the original original triangle ABC and that matrix of transformation it will be now our Elvis because it is transforming the image back to the object now let's see let's see what kind of mattress that you can find it through two ways are 3 ways the first way the first way of finding a matrix of transformation if given for example X has this one you can find X inverse by taking X X inverse can be found by taking identity or the X itself X this one x it inverse or the other way which is X inverse multiplied by X it should be giving you an identity matrix what is a identity matrix identity matrix is not done the one which gives you 1 0 0 1 and this one Maps identity matrix Maps an object back to its original position this is the first way you can find an inverse of a matrix by this method whereby you can now take X which is negative 1 0 0 1 you multiply by X actually which we don't know that is AP CT is the same as this one just taking this one backward and this one in and you should get 1 0 0 1 as an identity matrix and if you perform this way you shall get a you a value to be a this one will be negative a plus 0 which is that Y is cos to 1 and meaning that our air will be negative 1 because if you multiply you divide both sides by negative 1 you shall get our AP negative 1 and we now to get the value of P here you shall not buy this one with this one and you shall get negative 1 by P + 0 D so that is negative P is equal to 0 so many P is equal to 0 and now to find the value of C as always you take this 0 by a plus C is close to 0 so our C equals to 0 and and again we take 0 PI P plus 0 by 1 by T we get our T being 1 now we can say that we have found our a PCT to be a cost-effective 1 B 0 C 0 equal to a now we can replace in time in place of a we put negative 1 B we put 0 and C we put 0 and T is 1 we can find that the inverse of this matrix here negative 1 0 0 1 it is giving us the same as itself so meaning that if you transform this object from this from the piece but assess through a transformation X it will give you an image of a prime B Prime with these coordinates now we have found the matrix of that one egg X inverse is that one now that was our first way you can use to find a transformer the inverse of our transformation the second method of finding the inverse of a transformation it can be that if you are being given for example that let's say y equals to negative 2 0 0 negative 2 if given this one as you are transformational matrix you can still find the inverse of Y by using the palm tree method by finding first the determinant whereby the determinant is found by taking the leading diagonal or the main diagonal then you subtract it there at the diagonal whereby it will be negative 2 by negative 2 you get positive 4 minus 0 equals to R 0 0 by 0 you get 0 negative 2 by negative 2 you get 4 so 4 minus 0 you get our determinant equals to 4 now to find the in of why it simply means you find the first you taking one over the determinant which we have found here then here in the leading diagonal you exchange this little diagonal you exchange the values in the leading diagonal and then they add us the other the other diagonal this one like 0 0 if it is air or a B or AC you only change the signs if it is positive here then here you put the at the sign that is negative after changing their lit in the open all points and if it is negative here it will be positive for instant let's see what happens in in here we have negative 2 0 and we also have 0 negative 2 as our transformation matrix now the inverse we have found the determinant to before the inverse of this matrix will be 1 over the determinant which is 1 over 4 then in the late baroque you know we exchange we shall find if we exchange negative 2 to other it will be negative 2 and negative 2 to add a diagonal we find it will be negative 2 but this one it is still 0 even if we introduce negative 0 and here negative 0 it will still remain 0 now we find that that one shall remain that way so taking 1/4 x 4 you get a quarter by 4 you shall get this one gives you negative 1/2 and the other way it will give you 0 and this one gives you 0 and this is negative 1/2 so we have seen in case you have a transformation Y and any object being transformed using this one it can be mapped back or they American be mapped - its original for example if it is ABC triangle it can be brought from this a prime B prime C Prime and our Y by if you take this one which is y negative 1 when y is negative 1 this is a just inverse of Y which is our transformation and if you take the inverse of our transformation you multiply it by the image which is a prime B prime C prime we shall get our triangle back a PC that is our second method of finding the inverse of our transformation you just find the determinant after finding the determinant you take 1 over the determinant and then the inverse here you exchange the lipping diagonal term univille new variables and then the other diagonal you change the signs after that when you open the bracket and you find the inverse of your transformation the attachment that is whereby you have been given the vertices of those object and the image for example given a triangle ABC and then you have been given the image that is let's say a prime B prime C prime I'm just using that one to represent the coordinates now you can still move this one back this one represent vertices of which maybe it can't be 1 4 1 1 3 1 let's say for instant and negatives and then this one can be 1 4 1 1 3 1 you can still map this one back to this by an inverse of what mapped it back to that one what do I mean I in this way if the first triangle was part through a certain transformation we can still find an inverse of that transformation by mapping it the image back through the object whereby if you take the inverse you multiply by the object at the image it should give you the object vertices whereby now for our instant here we had a certain transformation it mapped it from this object to this image now we can find another transformation which can map it back and that one will be an inverse of the fast transformation if you take the inverse of this one then it will automatically give you the object meaning that if you take ABCD as let it be our transformation or our inverse transformation you multiply it by the odd the image which is negative 1 negative 1 4 negative 1 1 and negative 3 1 it should give you our object which is 1 1 3 4 1 1 what you will be finding here as the values of a b c d will just be the inverse of the original transformation i will be help you to pass through areas calcutta and determine ándara unlike as our six section infom from 2 we learned about how to find area scale factor and also how to find the determinant of a matrix in from 3 now we shall find a scale factor of our matrix even a certain matrix and then we see how the 2 relates area scale factor and the determinant in other way to find the determinant of a matrix we understood if even a b c d let's say this is our matrix m we can find the area the determinant of it by taking the leading diagonal which is ad minus C B this is how we said we can find the determinant of our mattress abbreviated as that now our area scale factor is simply taking taking the image area or area of the image over the object object area the image area which is the area of the image formed after a certain object has gone through a certain trans transformation and then the object area is that object its own area so if you divide the two you will find area scale factor and if you have a certain matrix area scale factor is just as the same as taking the determinant of that matrix determinant of cotton mattress but you find its absolute value what do I mean by saying absolute value absolute value of any given number is equal to the number without maybe let's say taking the only positive sign if for example a number is having a negative sign for example let's take negative 1 its absolute value it will be 1 and if inaba is to its absolute value we will still contain remain to be to the absolute value of any given number is just if the number has a negative sign you drop the negativity of that sign and you take the positiveness of that the number for example if it is negative 1 you drop the negative side then you take positive one if for example that is the determinant then what you will be getting that is our area scale factor for instance we can do a little practice on that by taking a matrix for example as negative 3 0 negative 0 negative 3 that is our first matrix and then we have another one like negative 1 0 negative 1 0 3 1 negative 2 1 1 0 1 1 then for example another one listen to 2 and 1/3 now let's say these are matrices but we have been told for example a certain triangle for example let's say triangle AP triangle ABC with the vertices a B our a 3 5b B 2 1 and our final vertice is 4 1 for example taking this one we can see that from this we can find the area of the object form area of object if you draw this one on a Cartesian plane of which here here it is 1 9 5 this is 4 so mean in the height of the triangle the vertices of a PC on Cartesian plane we shall get that our a b c give us a triangle of this kind whereby if you need to know the unis between this one you take this one is at point four this one at point two if you subtract you will get the distance from here tree is two units and the distance from the best this is isosceles triangle so the distance from the best to hide it will be one and here is five so if you subtract you get a it will be four four units the in-between we have four units so if we needed to get the area of this object now the area it is always 1/2 base times height but it will be 1/2 times our base here it will be 2 because from here to here is 2 but our height is from here to here which we have find from 1 to 5 we have four units so it will be by 4 and you get this one gives you 4 as unit for square units we get the area of the object be 4 square units and that one we write their ass for now area scale factor we can determine and also area of the image first they first determine the area of the image area scale factor and the determinant now we have found that if the object is having these vertices that is triangle ABC with vs. 3 5 2 1 and 4 1 we shall find that the area of the object will remain to be the same that is 4 4 4 4 but area of the image it will change according to which transformation is there triangle being subjected to if it will be subjected to this one we can still find the area scale factor and then the image whereby the area scale factor it will be determined by taking the absolute value of the determinant whereby now if we need to find the determinant of this one we can say we can get that negative 3 by negative 3 that is positive 9 subtract 0 you get the determinant is positive 9 so here it is positive 9 and here the determinant will be negative 1 by 1 you get negative 1 by 3 3 by 0 if you subtract the 2 you get negative 1 minus 0 it will give you negative 1 and the other one is negative 2 by 1 you get negative 2 minus 1 by 1 you get negative 2 minus 1 you get negative 3 2 by 3 you get 6 minus 1 by 2 you get negative 2 that is 6 minus 2 it will give you 4 now we have found the determinant and the area scale factor of the respective mattresses we shall find that it will be the area scale factor we say area scale factor is equal to the absolute value of the determinant and here our determinant for this case for the first matrix is 9 so the absolute value of 9 is still 9 I made the absolute value of 9 we're name is the determinant of the first matrix is equal to nine and that is our area scale factor the absolute value of the second matrix the determinant of the second Matrix which is negative 1 it will be equal to that is here now we shall find the negative 1 the absolute value of negative 1 and we get this one is equal to 1 and if we start to really find that the area scale factor of the second matrix is 1 and here negative 3 if you do the same key thing you will find here is 3 and here it will give you positive for any of the image now to find the area of the image we said earlier that area scale factor is obtained by taking the area of the image or image area divided by area of the object or object object area and for instance the first one we have 4 so if we have for our 4 is the area of the object to find the area of the image we shall place there like letter X for example here our area scale factor for the first matrix we have area scale factor b9 now our area of the image we don't have that is what we are looking for let's say let it be X now our area of the object area of the object we found for the hole the triangle the object is having 4 square unit if you do that way you you cross multiply you will find that our x equals 2 this one you will multiply by that this is x equals to 4 PI 9 you will get equal to 4 by 9 which is equal to 36 units square units so it's that six square unit the same applied the second one you placed in place of nine where there is Error scale factor you placed one and then where it is area of the object remained constant and that is four and you will find the area of the image here be four units I will add you to please continue filling the cell the third and fourth using the same procedure as before [Music] so we shall consider an exam unless this let's say we have point Q at this point and let's have another point here and I will see that if we transform it and a given matrix you find out that it would have the same so we are going to see that if you transform each other and given mattress you will find out that you will find an image of the first object being transformed as this we have here which is Trimble prime Z Prime and big crane with the points which were Q and M which was onto Q Prime and M frame on the another figure or image which we have obtained here so as we can notice here if this one was on a condition plane for example if this one was on a Cartesian plane you will identify that they all have the same fight this is up to four meaning inside is they the object ABCD is having height of Eddie which is equivalent to form unit from the same figure we can identify them as you see the image a bc l p NP prime L prime B prime C prime D prime is happy also height is a curriculum with a height of actually if you move that way you will find the height similar and the height is 4 units true so the height of that one now will be determined by you can take a tablet you can transfer that from this but you will understand that the height here actually the height is also 4 unit so meaning if both of them have the same height and we can see they're sharing the same base here that actually means if you find the area of LPC DD which is our figure here the area will be equal to the area of the image because they are sharing the same best [Music] they're sharing the same base at the same time they are having the same high so this vector for here is the same as the area of the parallelogram which we find is an image of what we have here and as you see the points moved with respect that and as you can see the line which is not far away this is the parent line DC is parallel to line AC and we can see it has moved to be D prime C prime but the lines which are not parallel lvd which are not parallel to our x axis or to our point here a B 2 maybe that language is not polish lately maybe now we have seen that it has intersected will they invent the image actually intersected with the length of the object so many that lines which are not lettuce which are not let me just say which are known peril I will use this importantly power to our invalid actually this one is not valid because if you view it well whether you will find out that and be an afferent beep remark on the same line something that is the invariant line so let us say let us which are not parallel to how invariant line intersects with the objects lines as you can witness opening that those lines which are much parallel to our event line of the image now these are the image slab that is a frame we can say a prime D Prime and B prime B prime C prime intersect without the object line intersects with the object as we can witness here therefore from here since we have said the independent line doesn't vary and also the area of these two figures which is the image and an object and the same therefore a matrix which Maps every CD this object to L prime B prime C Prime what we found here is what we call share mattress and therefore if you want to describe a shear modulus if you want to describe a shear matrix you will state the invariant line the variance line maybe on the y axis maybe on the x axis or maybe at the little but it may be identified as the X bar while lying so you have to stretch the invariant line after setting the Devonian plane you also state one of the point which is not on the in Valens name and we vary from one point to another for example here what I'm talking about if I initially used at one of the point which is noted on the invariance line is like point B which has been much to D Prime or C which has been moved to secret but those points which are on the divided line we see testing but on the same with wagon train so maybe they're not for you so this one I am referring to line the points which are not on the invariant line therefore you also state a point which is not on the variant on the valiant line for example if we could describe this our shear here we will now give our that this one is a shear matrix it's a shaman tricks actually let me just explain is a ship our tricks with X lying in the variant and then I have taken one of the pulley either D or C because they are the one which are mounted to the other so you take one of the point if it is d useless if coordinate then you state the coordinates of the image and you will have you will have described a sheer matrix fully so let's see an example Bobby has vanishes at O which is 0 0 B 3 0 and B is 2 0 and B is 3 4 so sure on my diagram its image and shear whose mattress is 1 0 2 1 let's see we have been told that we have a triangle a OB unless triangle has coordinates 0 0 for all we have 0 0 and it is having 3 0 and B is happy actually is 3 4 what we shall do because we have been told the ship named the shield transformed at Lycia we are being given us 1 0 to 1 so we need to find the image and instead we give out which kind when we describe that transform the transform of that ship fully for example we shall do this in order to find the image of that triangle we shall take the transform which we have we multiplied by the object coordinates then we shall find the image coordinates so for instant here we have the shear transform here will be given as 1 2 0 1 we are going to make like it by the object coordinates which we know for the all we have 0 0 and then for the air we have 3 H 3 0 and and for the B we have 3 4 therefore if we do that we are going to get the image the image coordinates for instead we shall have 0 0 4 if you do that way you will find if you much better you will get for the all this should be kept or ought not to look like 0 for a frame you will have 0 0 here shall give you 3 0 but the other one will you body and it will vary too if you must buy where you will find you're going to have 11 yeah now we have found all that we were given the object coordinates and now we have found out the image of the that object so we can give this triangle on a Cartesian plane and be able to describe its transform for example we are going now to draw a partition plane and mark the points on each and see which kind of a transform to place I am going to plot the points of the first triangle of which surround it here I am going to plot the points on the Cartesian plane and also plot the same points of the image on the same competition plane and see which kind of transform to place for example here we're going to plot and I find 4 0 0 here will be at that point for all and then air will be 3 0 so it will be at this point now that is for F and B is 3 4 so it should be somewhere here so you will find out that the triangle resembles rectangle triangle and this is our B so that is the travel we was the object now we have to prepare image I mean even you will find that the first point is the same as their object spoil so it'll remain here that is for Prime at that sister point and then for F prime is the same young which shall men but for betraying crouch and two level level form at this point but it is at 11 for it is at that point so if we know that travel you are going to get out you're gonna find out that this is not a big crime with the coordinates 11 so as we can see here our triangle a B as an ammonia mattress and it has become this new trend which we have aa prime V prime then B prime this is the odd image and this is the object from there we can see undisturbed a triangle and we can say our triangle is the matrix or the shear mattress which we were provided with was that she transformation to place with geometries of eggs invariant explain was invariant what that is that because we can see they are taking place on the eggs as the base so it doesn't wear this yes is invalid at the same time we have one point which is not only developed life but as much from B to B prime so we shall see is X invariant with with the B prime B of 3/4 as its content map also it has be mapped onto B prime which has 11 for man by that we shall have discovered that matrix because we can see if it is in the X invariant and one of the point map was from me which was referred to B prime which is Levin for the same can apply with the white variant and you only sent the invariant line and one of them which is not an invariant line but virus from one point to another so that is one once you stretch and give the coldness of that point let's see how straight matrix is also given out in general if you have been given a shear modulus then it will be if the x-axis is the invariant line then if the X is that environment let me just give axis environment because it may not necessarily be x axis if it's the in Valens name then you should you will have 1 K 0 then 1 this is the matrix or matrix with x axis invariant then if it is Y axis why-why-why lime environment or Y as the invariant line then the shear matrix which will be provided with now will be 1 0 K 1 this is our new mattresses which you may encounter and if you find one of them represented in that form then you know that 1 is a shear modulus let's see how we can find stretch we shall see one of the object in case we have this one yeah and you find that this object has been extended in such a way that if this one was a b c d it has now extended to form airframe the D prime B Prime and C frame if you witness you will find the points B has only moved perpendicular point a that is point B has moved perpendicular to point a and also C prime has moved perpendicular to point D which is an extract of point C with that we can also see that LD CB can be taken as often as our object and with that also you can find that the image in the mini mega form unified under given stretch matrix you will find that the image of a b c b is a prime B prime C Prime and then B Prime so with that we can say this stretch this object has undergone under a given stretch matrix so that it looks like it is elongated from their original position it's only an agate with points perpendicular today ones which are not valid so for example in this case if you want to give out here they we want to describe this stretch here you will say this is a stretch and if you want to be without you will now state also the environment line for our guest there if this one was grown to a scale or to a Cartesian plane you will find that if this was a Cartesian plane you will find out that this one was to the Y invalid at least we don't know at this point what that one is bad so you will find out that this one was at Y environment line was Y but the holes also moved with a different scale factor and if we want to find that scale factor you will take the image which moved that distance of the image then you divide by the original data for the object you define the scale factor for instance of this image here this diagram which is here if you want to find out the stretch scale factor here you will take a B prime M prime B Prime and M prime is the point of the new image form which is the the image now it will be R prime B prime which are the points of their image which formed then you divide by the original which is LV and if you find whatever you will be finding here is what we call scales back and after the it should be the same as the other than which value for example here will be D prime C prime of a B C which was the original lines and if we do that we have said this is what we is fun to scale factor if you have been given any kind of a matrix and if you draw it down then you should stretch the line which is invariant and also state the scale factor for example if given this point here and we have been told for example this one let's say it is abs we can extend this line and give it an example let's say I have a unit square here or as a unit square actually with the points ABCD as given below so this one may say after Bowie and a GUI are given stretch matrix you will find that it moves up to for example how to form then if I want to describe what kind of a matrix took place I will say this is a stretch matrix is a stretch matrix with as I can see here is the next line then the white line as is the white light which is invalid so I will say is a stretch mattress with it while I environment and you give out the scale factor of what okay for example there I will go to give up the scale factor and I've given you all you can claim you can calculate this girl factor for this instant we shall find the police which valid for example here we have V prime is C prime so the pulse which bar it is be varied from B to B prime but a and M prime of the points remains the same so we shall have if we need the scale factor we shall take L prime B prime which we have is from this point to that so as we can see actually from this point to there so it shall be the same as did not let us all that is fall it has moved with the four points so it shall before the original which is one point at the same which is there because if we witnessed here is just for one just for one so if we take the invariant we are going to take what a community so we shall get four by the original point which is from near PR or the same from here to here which is one so we shall find out that it has moved with a scale factor of four so we shall say with a scale factor so our unity circle and the when our stretch matrix with the Y lamb invariant and with a scale factor for so if you want to describe a matrix or stretch you give the invariant line and you also state the scale factor and if the scale factor is not provided you have seen how you can transmit from the image and objects which have been formed so in general if you want you want identify a stretch matrix in case you have X line invariant as the divided line then you will find your matrix will be in position of case 0 0 then 1 if you find a matrix which resemble that one you know that one is a mattress with s line invariant but with the scale factor KD that is the scale factor there and for the other one if Y line is dead by and time they just say invariant you will find out that you will have the opposite of that which is 0 0 0 then actually there should be 1 because that is the one vision listed so it's 1 0 0 K if you find this way you understand that is a stretch matrix with one main invariant and as a scale factor k with that we have come to the end of Shia and stretch mattress