 Okay, friends. So in this session, we are going to discuss one important topic determinants and We will see how these determinants are used to solve linear equations in multiple variables And we'll also see Kramer's tool and use of determinants in solving linear equations through Kramer's tools But before we start we must also understand what determinants is. So basically a determinant is an arrangement of arrangement of numbers Arrangement of Numbers in a particular grid format numbers in a in a square grid format square grid grid format and How is it represented? So we say let us say D is Equal to you put two vertical bars and write a B C and D like that. So if you see and then close it with another vertical bar, okay, so this is in our this is called our determinant determinant of Second order, okay, where if you see this is called These are called the rows So how many rows are there there are two rows in this case so here there are two rows isn't it and this is This one is called a column So there are two columns like that. Okay, so this one is column So you can see this is two cross two square grid Right. So how many elements are there individual members a b and c and d are called elements Right. So a b c and d are elements of this determinant Elements of this there are four numbers placed in that order. There is no Operation happening over here. So it's just plain plain and simple placement of four numbers right into cross to square grid structure Now what is the value of value of this determinant D? So value is written as debt DET and then within brackets you can write the The name of the determinant didn't be written as like that or it also expressed as D and Just like you find out absolute value. So you put two bars across D So this is that these are the two ways you can express the value of Determinant D and the value is nothing but a into D minus C into B. So how do we remember that? So Basically, this is the order. So you write top left right bottom multiply that and then minus Leftmost bottom into top right like that. So ad minus CD. So let's take an example and understand so for example if I have let's say D and D is let's say 341 and 2 Like that. So this will be equal to 3 into 2 minus 1 into 4 so it is nothing but 6 minus 4 which is Which is 2 Right 3 into 2 minus 1 into 4 2 another example. Let us take D is equal to minus 1 minus 7 here and 2 and 4 Okay Okay, and then a vertical line like that. So this is equal to what minus 1 So you'll have to take this multiplication first and then so hence it is minus 1 into minus 7 And then what you need to do you have to do this and subtract it from the original value So minus 2 4 into 2 Yeah, so this is 7 minus 8 and negative 1 this is the value of determinant of Second order Okay, there are few properties of determinant as well, but we'll take up in the next session and We also would be Taking up or let's say most importantly when this method of solving linear equations Using determinants the most important thing is when it is more than two variables, then you will also need determinants of third order determinants of third order So how do we get that or what does it mean? So it is nothing but simply this D is equal to and The elements will be as you could have guessed by now. It is 3 cross 3 grid 3 rows Three columns, right? So it is a1. Let's say a2 a3 then b1 then b2 b3 and c1 c2 c3 Okay, so now there are nine elements. How many elements three rows into three columns? nine elements, so how do we find the Values how we how do we find the values? Please pay attention This technique will be useful for you whenever you will be solving Any problem using determinants now how to find the value of D is simply this you take up any row or any column? So let us take the first row itself Yeah, you can take up any row or any column, but in in this case We are going to take up the first row. So let us say this is the first row. I am picking up Then what does what to do then you take the first element of the first row that is a1 You write it like that and then multiply it with now What you need to do is you cover the row and the column in which a1 exists So if you see if I have to cover this so I am covering the row and the column in which a1 exists So what is left over? Whatever is the left over thing you write that in the determinant form here So hence if you see B2 B3 C2 C3 is left after covering a1 So a1's row and column so hence it is B2 B3 C2 C3 Okay, and now but it's not all you have to also, you know Add a sign to this particular value. What is the sign? So you write minus one to the power row plus column number Okay, so of whom of a1. So if you see what is row number of a1? So R is one and column number is one. So hence R plus C will be Two right so hence I can replace this R plus C here and write one plus one one plus One okay now next next term is You have picked up the first row. So go to the next element of the first row So you write a2 and repeat the process repeat means you go to the second column of Sorry you go to the next element a2 of the first row right and write that element a2 now repeat the process repeat means what you cover the Row and column of a2 so the row is covered like that and then you cover the column like that and what is what is Free now. So if you see B1 C1 B3 C3 is free So write that as B1 B3 C1 C3 Okay, and then for the sign again, what will you do? You write minus one To the power row plus column. So row number is one and column number is two Right, and then again go to the third element of the first row in the row which you picked up So it is nothing but a3 and again. What do you need to do? You need to cover The row and the column again. So you cover the row of a3 and you cover the row of Sorry column of a3. So what is left? B1 B2 C1 C2, isn't it after covering the row and column of a3 so hence it will be simply B1 B2 C1 C2 and here it is again sign is minus one to the power row plus column of a3 So row number is one column number is three. Yeah correct, so this will be so this is the value of determinant D so there are three Small determinants coming up. Okay, this will be clear if we take an example Okay, so let us take an example to understand the calculation of third-order determinant. So here is a determinant given and One three four two one zero seven zero four. So how do we go about it? So to pick up one of the rows or columns. So let us say I picked up the first only So I picked up this first column and then you have to start with the first element. So let us let us pick up One right. So what would what would we write? We write one and then cover the Row and column containing one. So what is left? One zero. So she says this is left. Is it it? So what will I write? I will write within a determinant of one zero zero four, right? And what about the sign? So minus one to the power what? Row number of one and column number of one so one plus one first row first column Now come to the next term. How do I find out the next term? Take the second element of the first row which you picked up So let us say three and then cover three and the row of three and column of three So what is left if I cover the row of three? So this is row of three This is column of sorry. This is this was uh This one is the row and the other one. This one is the column Okay, so if you cover that what will be left you will be seeing it is two seven and zero four right and what about the sign minus one to the power Minus one to the power what first row and second column Now correct now third is take the third element four. What is the third element of the first row which we picked up? four So let us write four and then cover the row and the column of four So you'll see this one only will be left, right? So hence it is two one seven zero right and what about the sign minus one to the power first row third column Okay, now we know what a second order determinant is. So it is first first term is now minus one to the power two is positive one into one And then what is first determinant? value one into four minus zero into Zero, this is what we learned something back So first you have to do this and then this subtract this right Then the second term is minus one to the power three is minus one Into three and then what is the determinant value? two into four Why first you now go this way and then you go this way minus seven into zero Then third third term minus one to the power four This one is four. So minus one to the power four is one So plus one into four and two into zero Minus seven into one Okay So now it is simply mathematical Arithmetical Calculation, so let us do that So one into one so one into One into four is four and minus zero into zero is zero Then minus one into three into eight Right and the seven into zero is zero and then one into four Into two into zero is gone. So it is minus seven Into one Right. So it is nothing but four minus 24 and then minus 28 So it comes out to be minus 48 So this is the value of the determinant Of third order d Why are we doing this because uh in the next session we'll learn how to use determinants To solve linear equations in multiple variables Especially if we are using three cross three determinant Then we will be using this in solving linear equations in three variables You can have determinants of fourth order as well So determinant of fourth order will look like something like this. So You have four rows and four columns right one two three seven nine minus one four two Five minus six two one one zero zero one, right? So this will be Determinant of fourth order will be used in solving. Let's say linear equations in four variables Linear equations in four variables now you'll ask Where are these equations used? So, you know in higher computations where Usually in statistics and many other application areas Where we have lots of variables and lots of linear equations There we have to use these particular method only To solve linear equations Because otherwise the the conventional substitution and elimination Methods will become very cumbersome So I hope you understood the determinants and how to find out the value of the determinant There are many other things related to determinants which we'll come across when we start doing determinants This this small session was only to introduce determinants to you so that you can use Kramer rules for solving linear equations in multiple variables Thank you