Business Mathematics and Statistics Subtitles PDF

I clearly audible to all I audible to all. Alright, very good. So everyone post your UIDs with all your details and then we shall start in the meanwhile. I'm sharing my screen. Good evening everyone. Good evening. Uh, did you attend the questions which I have posted yesterday on the discussion...

I clearly audible to all I audible to all. Alright, very good. So everyone post your UIDs with all your details and then we shall start in the meanwhile. I'm sharing my screen. Good evening everyone. Good evening. Uh, did you attend the questions which I have posted yesterday on the discussion forum? Everyone has done those questions, those were easy. Could you, were you able to attend those questions or you have any doubt? Okay, fine then let's continue. Today we'll be starting with the new chapter that is matrices and I believe, okay, as I'll see you post on the discussion forum. I'll post the solutions there, okay? Because here we'll complete our lecture and then we'll take your queries on the discussion forum. Okay? You post there, I'll post my reply there. Here. Let's not waste time here because metrics is a one full place chapter. We have to complete that first. So I'm sharing my screen. You'll let me know if it is visible. It's visible, everyone. Alright, this perfect. Okay, so yesterday we have done this. I'll share these notes with you and uh, did you make your group, anyone, any one person you please post your number here and you create your WhatsApp group who will take this opportunity to create a group? Anyone? Her Simran? Yes, I am sharing my number. This is my number. You create your group, add me also there and we can take your queries problems there. Okay, everyone share your phone numbers here or you give, you create one group and you post the link here, make a group and you put the link here in the WhatsApp group here in the chat boss so that everyone can join. Okay? Right now, yes, you please post your link right here so that everyone can join currently. So let's start. Today we are starting with our new chapter, which is mattresses and I believe you must have studied earlier also. Everyone, everyone please let me know. Do you have any idea about mattresses? Yes or no? You must have studied in 10th or plus one, plus two. Yes or no? Let me know. What are mattresses? Suppose I have equation two x minus three Y equals to seven and five fits plus four Y equals to 11. This is my one set of equations. Another left is something like seven y plus three Z equals to eight. One more is three plus zero by six. That equals to minus 11. Suppose I have these two equations, these two set of equations. This is one, this is second and now I want a solution for this. How I would be valuing the solution for this. Do you have any idea? Tell me Rectangular area of numbers are Indian row. Uh, very good, yes, so this is the technical definition of the matrices. What do you understand? If I'm asking, I want the solution for these two equations. Yes, very good. Now everyone join this link. Okay, now what do you understand? You see mattresses hast proposed in the year 1857 by Arthur Kali. Arthur Ka, okay, he has given this idea knowing the value of variables. Yes, ika. What is happening here when I have a set of equations in these equation, what you observe, I have two things. One are the values which are 2, 3, 5, 4, 7, 11, another. These are what? These are my proficiency. Proficiency to what? Proficiency. Two. Some variable. What are my variable here? X and RY. My unknown variables, these are my unknowns, which are known as variables, which means it is going to assume some values. Now these coefficient are connected to my right hand side of the equation and which is seven and 11. By sub relation, what is that relation? This relation is equality, which is making it an equation. Now employees, the set of data which I have on my left hand side, what is that set of data? Oh five minus three and four with respect to my unknowns, to the right hand side is my seven 11. So this is my set of data, this is my set of data. Set of data. This is again set of data. It's telling if I assume certain values for xy, then my left hand side data is all going to be equal to my right hand side data. Similar manner for my second set. Also, I would be having my set of data as two three minus 7 0 3 6. This is going to be equal to eight minus 11. If I have the value of my unknowns, which are my unknowns, X, Y, Z. So these are going to be equivalent if I get to know the value of X and Y. How? Now the point is how we are going to evaluate the value of X and Y in order to get the value of x and Y. This theory of mattresses were proposed by Arthur Kelly in 1857. He said if I have any set of data which can be arranged, set of data, which can be arranged in a manner that makes some sense in arranged in a manner and what is that manner in an order M, cross and order where my M are number of rows and NR number of columns and cross means it should assume that structure and what is that structure when my number of rows whatever are my number of rows. These are with respect to the number of columns. It means these are going to be my rows and my N are assuming these columns. So these are, these are my mattresses when I, it means this is order of an A. So what is my order of Aaron which is M cross M. So when I know how to write the Es, there are so many types of Es, there are so many types of mattresses. Can you name any? So these I have written in the shortcut. We have so many types of mattresses here, square mattresses, we'll take one by one. Okay, so it is telling see singular, yes. We'll take one by one. So here it is row. Now first is row. So every one you don't have to learn definitions. I will just giving you one example for every metrics you will get to know, I have written so many metricses here, nearly 27, I don't know how much you know and you don't have to learn on, just know the procedure, how to evaluate or how to use these metricses. So first I say metrices, you know it would be written in the order of cro N. So if my order is m crossen and I'm telling it is a raw metrics. So raw metrics means whatever is my role, it is going to assume the value one. But I can have number of columns. So if it is happening in this manner, then my metrics is say suppose these are my elements. 4, 5, 6 and so on. So my row number is one, but how many columns I have, I have considered here six columns and these are my elements. So whatever are the elements in the metrics, these are known as elements and whenever we are representing a metrics, metrics is always represented in uppercase and the elements, elements of metrics are represented of metrics are represented in lowercase always. So if this is my metrics, R one, so capital R one is my metrics, my elements here are 1, 2, 3, 4, 5, 6. Now you'll say these are, these are numeric values. How I can represent this as lowercase, these are assumed by the position. This position is represented in the manner A IJ, which takes the order and process. So it means my metrics. R one is written as metrics of a IG in the order M process employees here, here my I is one, my JR six employees. Now you substitute the value. When I substitute, what would I have my, I would always remain one because I'm having only one rule J I'm having six. So this means my first J, this means what I'm going from 1, 2, 3, 4, 5, and six. So my second element would be what again it is my one rule only. Now my column is separate or different. So similar manner third, fourth, fifth, and sixth implies that is my A one one, it is one my A, one, two. This is two A, 1, 3, 1, 4, 4 A, 1, 5, 5, A 1, 6, 6 clear. So this is the way we represent the metrics. So we have done the row metrics. So similar manner we'll have the column metrics. So in the column, which means if I have R one is my metrics or let me take it some common name. Suppose my metrics is capital A. Now I have a column metrics. It means again in the M cross end my column is one but I can have number of rows. So my rows are M, so it is M cross one. So this would be M cross one. So it means I have row one, row two, I have row three and I have only one column and these are my elements. Similar manner, the element in the terms of notation. When I represent this, what are, what is going to be the value here my row are three total number of three which are assuming the positions one to three, but my J it is only one. So when I substitute it, how would I write it? My row is one, column is one, my row is two, column is one, my row is three. But column is one where the position of a one, one takes the value, one element a two, one takes the value two element A three one takes the value three. So this is the way we write it. Alright, so everybody knows those who don't know. So this is the way we write metrics, okay? Now it is square. So I believe now by now, if you know the row metrics, you know the column metrics, then you know how I can write it. Square, square means what? Everyone knows square. So if it has the value, A my all the side will have the value A only E. So that is why if it is a square metrics in place, the order of my metrics, it would be same. It means my row would be equals two, my number of columns here, then it'll give me a scale metric. For example, my metrics say has the value 2, 3, 4, 6, how many number of rows I have two, how many number of columns I have too. So this makes my scare metrics. Then you have diagonal. What do you understand by the diagonal metrics? Diagonal means if this is my scare, my diagonal is this. So whatever is the diagonal, I have elements only in the diagonal. So if my metrics is supposed 2, 3, 4, 6, so I should have only the diagonal elements implies I have here two six. But instead of these value I have zero. So in short, good diagonal will represent like this. So if I write two and six, if I write diagonal equals two square metrics or these are bracket, uh, I forgot to tell you when we are represented metrics we always take the structure of this square bracket. It might be also written as the parenthesis, okay? Both are acceptable. So by default it is here. I can also write in the route bracket also. So if it is written square bracket two six, it means these are the elements in their diagonal rest, all the elements are zero, okay? Both ways it is written scale scaler metrics, what do you understand by scaler? Scaler means any non-zero number, any non-zero real number and by default we represented by the symbol lambda. Everybody knows this is lambda. Those who don't know it is some Greek value, we represent it as lambda. I can also represent it with the new delta but by default we assume this as lambda. Okay? Now if I write my metrics A is 2, 3, 4, 6, I want the value of two A. So here my lambda is two because this is a number which is a non-zero value. Now I want the value of two A, it means two and two metrics. A how you'll evaluate my this two is going to be multiplied with every single element in the metrics. When you multiply it, what you will have two into 2, 4, 2 into 3 6, 2 into 4 8, 2 into six 12. This is the value of two eight. This is known as scaler metrics or scaler multiplication A metrics. So scaler metrics means you will always have some s scaler that has to be multiplied with the metrics itself. Unit or identity unit metrics or identity metrics. So unit or identity unit means everybody knows unit would be one. Yes. So in metrics we represent unit metrics with capital I, okay it means I will have the value one where only in the diner if I have such kind of a metrics, this is known as unit metrics or identity metrics. What is the order of metrics to cross two? So if it's order of two, I can also represent this as I two. So if it is written I two, it means it is of order two. Similarly, suppose if I write I I four it means I will be having four rows, I will be having four columns. Another thing is identity metrics is always a square metrics, it is a scare metrics and it is also a diagonal metrics. We can say no it is a scare. Identity will always be a scare metrics. Just remember diagonal it is only having the elements on the diagonal. So if I have written I four it means 1, 2, 3 and four, right? So these are going to make my diagonal. So what are other values? 1, 2, 3. So this is second position, this is third position and this is my fourth position. 0, 0, 0 1. So in short it is written as capital i four understood everyone. So this is the unit metrics or the identity team. This is also known as zero. Alright, next is zero. Zero metrics, null metrics, empty metrics. It means I don't have any element here. So every single element in my metrics would be zero. Would be zero. Suppose I'm writing this. So what is the order of my metrics? Two rows and three columns. And in shortcut this metrics is represented by capital low. This is upper case of O, this is upper case of O oh I. Okay O. Now next is symmetric. Now symmetric and skews symmetric are very, very important definitions and you will get many questions based on this. Just a minute symmetric and one how you write it symmetric. What is the symmetric metrics? If you have studied earlier, what do you understand by the term symmetric? Someone is saying anything one symmetric metrics. Do you not know this is not right. It means if I have a matrix, this is equals to transpose. So let me tell you first what is transpose? Because this symbol which I have written, this means transpose metrics. Transpose methods, okay, now transpose means opposite if I have a row, this becomes column. If you have a column, if you have column, column becomes row, it's not writing. So just understand row is column. Column becomes row implies. If my metrics is two three, if my metrics is 2, 3, 4, 6, this is my sum metrics and I wanted transpose. So it's transpose would be what are my rows here my row is twofold, which is my first row. So first row will become my first column. Now similarly, my second row, this will become the second column. So this is the transpose. So if you take the transpose of the transpose, if I take transpose of the transpose, this is going to give me metrics again how you check again here. So I want to take the transpose of this metrics, what is my row? Two three is my row. So whatever is row, it becomes the column. So this is my column. Now second row becomes your second column. So what you observe my this metrics is again equals to a only hence proof. Now symmetric metrics is telling whatever is your metrics. If this equals to the transpose of the metrics then this is known as symmetric metrics. What is going to be qmetric Q symmetric Q symmetric. It means whatever is the metrics, you take the transpose of that metrics, this should be equals to minus C. If this is happening then it is set to be skew symmetric metrics. Okay? And let me show you here because I want to cover all these fast. See it is clearly visible to all. There are certain properties also we'll come to the operations later. Let take you here. Transpose of the metrics is always the transport. These are the properties of transpose I have just told you transpose. So if you take the transpose of the metrics and you again take the transport, you will get the metrics itself. Similarly, it is true for the addition of the metrics you add to metrics and you take the transpose or you take two transpose metrics and then you add you'll always get the same back and in the transpose if the product is taken then you take the transpose then or you take two transposes and check the product. Values are always made after this symmetric metrics. So if the transpose is equals to the metrics, this is going to be symmetric. There is one rule that is whatever is the position of the element on the diagonal. These are always going to be seen. That is a IJ position will always be equals to a JI position in this symmetric metrics. We'll do the numerical. You will understand SKU symmetric. It means if the transpose of metrics is equals to minus a. Now here is again one condition that in the symmetric what is happening A IJ position will be equals to a GI in the sku symmetric one will be opposite. That is negative. Either this is minus A or a IG equals to minus a GI both may be true. Alright, so in a metrics either this will hold or this will hold. Plus there is one more condition that is your diagonal elements are always zero in these qmetric metrics. Intary it means if I take a metrics and I take the product of that metrics with itself, you will always get identity metrics if this is happening. This is known as intary metrics, alright? This is known as intary metrics, singular metrics. I'll tell you a joint later. Singular means determinant of the metrics will be zero non singular. Determinant is not zero and determinant we will do tomorrow. So you I will tell you this clearly tomorrow inverse of the metrics inverse formula is this and for this I require a joint. Now what is a joint of the metrics? Now you see here, and I guess you must have studied a joint IND plus two, a joint metrics. So a joint metrics for this you should know what is, what are your minors, what are your minors? Then you should know the cofactors, whatever is my metrics. Suppose I have 2, 3, 4, 6. This is my metrics. A I want minors how many miners I will have the number of elements you have in your metrics. Number of miners I will have, so I have and minor are always represented with capital MI, ig four factors are represented with capital A. Okay, so let why this metrics is capital X right now I want minors, how many minors I have M1, one capital M, one, two, capital M two one, capital M, two, two. So I'll have four miners. Similarly, number of co factors, a 1, 1, 1, 2, 8, 2, 1, 8, 2, 2. These are going to be my number of co factors. After this you have to change the sign. Since I have to cross two metrics, it'll always assume these sign, if you have three, cross three metrics, it takes the value of the sign like these plus minus, plus minus means. Now you see here carefully I want value of M1 one. You cut the row first row, first column, whatever is left. This is my minor. Understand now I want one, two, it means you cut, cut first row and you cut second column. What is left? Four is my, this is my second minor. Now I want two one it means you cut second row first column. What is left? Three is my third miner I require next. So M two two cut the second row, second column, what is left two. So the metrics of minor is first element is 6, 4, 3 and two. This is the metrics of the minor. Now I require cofactor. Cofactor means you just change the value of sign, you change the values of sign. So this is my metrics of minor. You now change the value of minors. First is always positive, second is negative, then it is negative. Then again it is positive. Now after this you take the transpose, you take the transpose of this methods. So when I take the transpose I will have the A joint six first row will become the column. Now six minus four, minus three and two. So this is your metrics of the A joint. So my earlier metrics was 2, 3, 4 and if it's a two cross two metrics, I'll tell you the shortcut. Now listen very carefully, okay, this was my earlier metrics too. It is clearly visible. No, 2, 3, 4, 6 is my metrics XI want the A joint of this metrics. So a joint of X, it is only true this holds true when I have a metrics of order to cross two. Okay, for three, cross three or for the higher order it is not applicable. So if you have to cross two and the A joint is off, I can do directly. This is my earlier given metrics. 2, 3, 4, 6. You interchange the diagonal element. What are my diagonal element two and six. So you interchange it. What are the elements of non diagonal three and four? You just put the minus there. So I will have minus three, minus four. Now what you observe, you are this metrics and you this metrics, they are exactly same and it'll always be same. Understood. This is the way we find the a joint of the metrics. It is clear to everyone how to find the ajo of metrics. Say yes or no, especially the non-med students. It is clear how to find their joint. A joint means it is a kind of uh, explain again. Alright, I'm repeating again. Listen carefully. I have told you two methods. One is the shortcut, one is the long one, okay, I'm repeating again the long one first listen carefully. Earlier, my metrics is given as 2, 3, 4, 6. First I require minors. So these are your minors, M-I-N-O-R-S. Then you require core factors. Okay, four factors. So minors are represented by capital MIG minus means first whatever is your metrics. Accordingly you write the position of the elements. So my two is one, one position three is one, two, position four is two, one position six is two. Two position, right? The first you write in the rough your elements, then you write it of according to the position one. One means you cut the first row, first column. So when I cut the first I, let me take this column. So when I'm cutting first row, first column, what is left six is left. So this is my first element. Now I have to cut first row, second column. The first row, second column I will cut what is left? Four is left two, one cut the second row. First column, what is left? Three is left two, two it means cut the second row, second column, what is left? Two is left. Now according to whatever miners you have, you write in the metrics. So you will have the metrics of the miners. Now I want core factors. Core factors. Me, I just have to change the sign. So if you have two cross two metrics. So you just remember the sign bus, you just put the sign. So I have first positive, then negative, negative positive. So I have written positive, then negative, then negative, then positive other. I have three cross free metrics. So this is the order of the sign. Now when you will have your co-factors, this is the transpose means row becomes the column. Column becomes the row. So six and minus four is my first row. This will become my first column. Now minus three two is my second row. This will become my column two. I have whatever I have. This is my A joint methods. This is the joint metrics clear. Now it is clear the longer method everyone. Alright now shortcut is, now listen very carefully the shortcut metrics X was 2, 3, 4, 6. This is my original metrics. I want its shortcut and this is applicable only when I have to cross two metrics or so. Two cross two may, whatever are your diner elements opposite its this would be six two now and non mapper just put the minus understand Abby match a joint matrix score. It is exactly same. Clear as a poor how to find the joint. Alright? If it is clear, you gimme the answer for this two, find the A joint, give me the answer shortcut. Find a joint of effect. Tell me fast everyone I want to complete this. William come to the operations, give me the answer. What is the answer here? Gimme the answer first. Nobody's responding. What is the answer for this two? Uh, CI want a joint no gon elements opposite kernel. So my diagonal elements are minus one and two, ER two is here, minus one is here or whatever are left. You put the minus there. So I have minus nine, minus 13. This is my answer. Understood? Now, yes or no everyone I'm left with 20 minutes. Alright. Okay, Parran. No not understood. Parran, you did not understood one note is not working to place nine and 13. No I don't have to replace that. Her simran only diagonal malus has to be exchanged but she listen again. C, what are my sign? Positive, positive, negative, negative. Right? So metrics, apana metrics was 2, 3, 4, 6, 2 6. Co U will ex exchange what? It becomes six and two, three and four per I have to put minus sign. Okay, now see what is my metrics here? Minus one and two. So this has to be exchanged. So this becomes two and minus one. Now what now? Now I have two change the sign of the other non diagonal elements both are positive. Here I have to make them minus understood. Now, yes or no, this was very easy. Whatever are your diagonal elements, you exchange them and whatever are the different elements just changed the sign here it was both the negatives, uh, both were positive. So I just have to put the negative after you take one, another example, now I have minus one, minus two, four and zero. This is my metrics. Now you tell me the answer. Now you tell me the answer. A joint topics 20 minutes left. I have to complete more. You tell me the answer for this. These are your metrics. Now I want the A joint. Tell me ika and hard simran tell me fast. Hurry, hurry up, hurry I have to complete. Please hurry up. Zero ika is correct. Tik, you again made mistake. My diagonal elements are these exchange it. Yes, yes, Rakesh is correct and non diagonal elements are these two N try sign would be changed negative to minus two. Ka again you take minus. So this becomes positive. This is my minus clear Now so this is your A joint and if I do using another method means the longer method. So what you will have uh, yes it is clear now. So this is our method one. Let me take the another method using the minors. So first I want minors. For minors I require M1, 1, 1, 2, 2, 1, 2, 2. It means first row, first column gone. What is left? Uh, I have cut the wrong one. Add join first row first column one. So I'm left with zero. Then first row, no first row, second column, first row, second column. I will be left with four. First row, second column four, second row, first column minus two, second row, second column minus one. So I have minus metrics. Now I want cofactor me. So in cofactor what you do, we only change the sign positive, negative, negative. So this is positive now and positive. Then you take transpose. So my row becomes column and the row becomes next column. Now you check your answer, the shortcut one. So this is why method two, this is my method one. What is the answer? Both are matching. Both are matching clear longer method. Why? Why minus one in a minus one, minus one minus two, four and zero to minus one to interchange yoga. Understood the safe non or or understood? I'm telling you again I have to complete my syllabus here. Metrics. My metrics here is I'm writing again here. Minus one, minus two, four and zero. The minus one one first row, first column zero idea first row two. The first row, second column four is left, then second row, first column gone. Minus two is left. Now second row, second column gone. Minus one is left minus. Now ign change the sign goes like this. Positive, negative, negative, positive. So whatever metrics I have, give all metrics, I'm changing sign change Camila. Positive, negative, negative three. Negative. Yeah, negative. Negative. This is positive Now and this is positive only positive, positive, positive is positive. Positive. Negative is negative. Negative positive is negative and negative. Negative is positive. I guess what we are getting minus one kaine SA change minus one to minus one here. Understood. Remember this? I got it now you were confusing with this clear. So this is the way we valued the A joint I believe for your syllabus as per your standard only this much is required. Now we are moving to further operations. So a comparable metrics means the the number of rows and number of columns should be equal. Equal metrics means if I have two metrics, both the orders will be same. Also suppose I have metrics A and metrics B. If my metrics A is of three, cross three. So my metric B should also be of three, cross three order another thing, whatever are the elements in a I should have same elements in B also then this makes your equal metrics. This we have done row column, zero, square diagonal, then scaler. I told you identity I have told you hard triangular was left triangular means uh, you know what is the triangular metrics? So you see observe here, this is my triangular metrics, alright? And this is also my triangular metrics. These are two different types of tri. When I have numbers in the above, this makes upper triangular. When I have elements in the lower triangle, this makes my lower triangular metrics and my diagonal elements not necessary has to be zero. It may be non-zero, it may be non-zero. Alright, so if I have element, suppose I have metrics like this. 1, 1, 1, 0, 0, 0, 0, 0, 0. Now what do you observe? I have elements here only. So this is again a triangular metrics. It is both upper triangular also it is lower triangular also. But if I write here, suppose if I write here zero, one and zero. Now what you observe now I have here non-zero element one. So this is now it is a lower triangular matrix. It is not the upper triangular metrics. Now understood wherever I'll have element below the triangle, below the diagonal or above the diagonal. That is going to be a triangular matrix. So I may, I'm having say, suppose have two. So when I have two below the diagonal, this will be lower triangle. And suppose I have another metrics. Suppose it is like 1, 0, 0, 2, 0, 0, 0, 0, 0. What do you observe? One row, 2, 3, 1 column, second column, third column. So this is three cross three. My diagonal is this. Below the diagonal all are zero tk, but above the diagonal I have this two. So it means this is my upper triangular metrics. Understood everyone. The difference between upper and the lower triangle. Alright, so these are the questions you try yourself. Addition, I tell you addition means you have two mattresses. Whenever I want to add these two mattresses, it always are, these are added on their positions. Two will be added to two will be added to this position. It means you always write in the terms of elements. How many elements? I have a 1, 1, 1, 2, 1, 3, A, 2, 1, 2, 2, 2, 3. In metrics B. What I have B one one B, one two, B one three, B two one, B two two, B two, three. It means. Now I want to add these one. One will always be added to 1, 1, 1. Two will always be added to 1, 2, 1 3 will always be added to one three. So two one will be added to two. 1, 2, 2, 2, 2 2 3 with two three, understood similar manner. If I'm adding similar manner, if I'm subtracting also again one, one will be subtracted to 1, 1, 1, 2 with 1, 2, 1, 3 with one, three and so on. Understood everyone. So now you look here in this example, two is added to four, this is six three added to four, uh three with a nine, 12 minus one plus minus 11. So this is negative, negative. This is minus one. Plus and minus is again minus. So this has become minus 12 ika. Understood This sign, no how we are heading minus one plus minus 11. So this is minus one, plus minus is again minus. So this is minus 12 five plus 3 8, 0 plus 1, 1 7 plus minus two. This is seven minus two. So I have five. Understood. This is the way we do the addition. Now, subtraction. Now this is my metrics A, this is my metrics. BI need to subtract. So four minus two. That is two minus seven minus four. So this is minus 11, minus two, minus minus five. So this is minus two minus minuses plus. So this becomes three now five minus, minus nine minus minuses again plus five and 9 14 3 minus six. So this is minus three zero minus minus three. So this is positive, understood the subtraction everyone. Alright now these are the properties. Addition is always competitive. Competitive. Wherever the word competitive is written, it means A plus B will be always equals to B plus A. For example, you write, uh, everyone is having a notebook. Now everyone is having a notebook, yes or no. Yesterday I told you no you should have a notebook. So if you have notebooks, you write solve this example, not now, but after the class, this is my A and this is my b, B is 4, 9, 3, 1 ticket. It's not clear. Let me write it clearly. My B here is four, nine, let it be minus four. Six ticket A is 2, 1 0 3 B is minus 4, 9 0 6. You have to find the value of A plus B and B plus C. You tell me it is correct or not after the class, okay, solve this. So this is your portion number one. Now addition is always associated. Similarly, ab, I have given you, you take the value of C as this four and 5, 1, 3, 4, 5. This is your C. Then you find the value of this everyone okay? Not it down. I have given you the value of A 2, 1, 0, 3 value of B is minus 4, 9, 0 6 and your capital C is 1, 3, 4, 5. You have to prove this law, whatever it is telling it is true. This means A plus B in bracket it means first you add A plus B and then you add here C metrics. Understand the similar manner on the right hand side. First you add B plus C and whatever answer you will have usme, you will add a. Now you check your left hand side car value, right hand side car value, it is equal or not understood. Similarly, existence additive identity. So whenever I have metrics say is may. If I add capital O, what is capital O? Capital O is all metrics, right? So whatever is the metrics is may suppose my metrics is 1, 0, 1 and two. Here I will add Abbi, whatever is the order of my metrics. Same will be the order of my null metrics also. So if I have taken two cross two, I have to take the null metrics also. Two cross two Abbi, you add this, you will always have metrics. A only IGA one may add 0, 0, 0 plus 0 0 1 plus 0 1 1 plus zero P is one and two plus zero. This is two. So become familiar like this metrics and this metrics both are same. No. So this is what it is telling either I find a plus zero or zero plus and this is not zero. This is null metrics understand and I told you no null metrics we represent with capital O. So this is capital O, this is capital A, this is capital O. Capital O means null metrics and whatever will be the order of metrics say that will be the order of your metrics of null metrics. Also understand the similarly additive inverse. Inverse means inverse opposite casa opposite. If I am having a positive metrics, those opposite would be a negative metrics. Understand inverse metrics, inverse metrics. The formula is this a joint overdetermined. This I will tell you tomorrow when we'll do the determinant. That is why I have not told you the inverse methods. Abbi, you just remember inverse metrics. The formula is a joint of metrics upon the determinant of metrics clear. Now I require additive inverse. Additive inverse methods. Suppose I have four, Abby I add here minus four. What you will have tell me if I have four and I am adding minus four, I'll have zero No. So this is if I get zero after adding something, then that is known as additive inverse. Understood everyone, whatever is the number you have to add quantity. So additive inverse understood to minus four. Here is additive inverse clear. Now here example is given metrics A is given a B, he is asking you find the additive inverse. Additive inverse two is written minus two minus three zero ANA plus three zero to zero. The zero may I have two zero, add zero. Similarly I have five. I have to make it zero. What I will do, I need to add minus five. Similarly, I have plus seven. I have to make it zero. I have to add minus seven. Similarly, I have my have minus six. I have to make it zero. I have to add positive six. So my this metrics is the additive inverse of this given metrics understood everyone how you find the additive inverse. Now let me take the, we are 11 two minutes. Let's do the multiplication. So these were the operation of the metrices. Now everyone knows how to do multiplication. So scaler, multiplication. I have told you in this start, if I have a metrics three is known as my scaler, you multiply it, you will have a scaler metrics. So these are the laws of scaler metrics. That is whatever is the element scaler. You multiply, you'll get T, you have to multiply individually with each metrics. These are the law will do the question. You will understand I want to do the product multiplication of metrics. Now you see here one and I have metrics. Say I have metrics. P, everyone know how to do the multiplication. Just remember I'm left with two minutes. Pay attention. Okay, how many rows I have? Row one, row two, column one, column two, multiplication is possible abi. Both are scale metrics. No, both are scared. So always remember my, this position should always be equal to this position. It means whatever element I have here, I should be the element here. What is my two here? This is my row. Multiply with the column. What is row here? Two. What is the column? This one, it means column of my first matrix should be equals to the row of my second matrix. Then only multiplication is true, otherwise it is not true. A b, this is equal square metrics. So obviously I can do it. Now look here I have two rows. I have three columns. Now look here I have three rows, I have two columns. So my number of rows, columns of first metrics is equals to my number of rows of second. It means I multiplication is multiplication is going to hold error and whatever will be my answer of the metrics. SCA order will be two, cross two SCA order will always be two. Cross two. Understand. Now suppose now suppose I have the metrics minus 2 1 3. This is my same only, and my metrics B is 2, 4, 9 5. Now what you observe how many rows I have two. How many columns I have three here. Number of rows are two, number of columns are three. Now three is not equals to two. It means I cannot multiply these two metrics. My metrics says this. My metrics B is this. Then I cannot multiply these metrics. Understood everyone. Multiplication is done in this manner. The processor, I will tell tomorrow the class is over. If you have another class, you have to join it. If you don't have, then I can tell you multiplication also. You have another class right now everyone? No, no class. So if I, I take only two minutes, we can do the multiplication. Tell me first. Otherwise I close. Yes. Alright, continue. We'll do the multiplication. Okay.