 Now we've had a very nice gentle introduction to matrices and vectors and we started off even looking at these systems of linear equations Now let's just go one level deeper I think you are quite familiar with it now you're comfortable with what a matrix is what a vector is How some matrices and augmented matrices represent systems of linear equations? Let's go deeper and the topic for today is this concept of an identity matrix Now you are quite familiar with what it is because you know what an identity element is if I were to look at the set of all Integers if I were to look at the set of all integers and I would look at the binary operation of addition I don't worry about those terms if you've taken a course on Group theory or abstract algebra and there is a playlist are linked to it There then you will know what this is all about What it is to saying though if I take all the if I take all the integers So that would be this set Say minus three minus two minus one zero one two three All the way to infinite negative infinity positive infinity and my binary operation is positive binary Just means I'm taking two elements and I'm adding them together So I'm going to take one plus two one plus three negative one plus three So take two of them and add them to each other It's as simple as that don't let these learn that the jargon and mathematics ever ever ever get you down It's a simple stuff. I'm just adding two of these There is an identity element for addition and that is zero because it works like this if I take any element here Three plus zero that equals zero plus three and that's a three Identity means if I do something to two elements in the set. I'm just left with The if I do something of what with one of the elements with identity element I'm just left with that element if I look at the integers and I look at multiplication The identity element is one because one times any of these elements So one times three that equals three times one and that is just going to be equal to Three again So that is just the identity element leaving something the way that it is supposed to be or the way that it originally was and There is this concept of the identity matrix Now let's just very quickly think of the identity matrix as far as addition is concerned So I have this matrix two one three four and I wanted to add Some matrix to it so that I'm left with this original matrix Well, what is the identity element for addition going to be zero zero zero zero we've looked at matrix addition It's element wise two plus zero one plus zero three plus zero four plus zero and I have it right there two one three four That's the identity element. It's nothing other than this identity element. So no problem there What is the identity element though for multiplication? What is the identity element for multiplication and that's very simple and we are restricting ourselves here By the way to square matrices number of rows equals number of columns only those an Identity element is very simple one and zero and zero and one It is as simple as that if you take any matrix and you do this multiplication. Let's do that Two and one and three and four Let's do one and zero and zero and one as far as my multiplication is concerned. So So for this one, it'll be this row this column and this row. So two times Two times one is two plus zero. That's two for this one zero one for this one three Three and for this one three four and lo and behold I'm left left with exactly the same thing what you wouldn't notice. Well, just by the way, we call this identity We call this identity matrix I and we know we can scale up as long as there's ones here and zeros everywhere else So that would be the identity element. That's three by three. By the way, this is called the main diagonal The main diagonal of a square matrix Top left to bottom right if they all ones ones everything else is there That is an identity element and if this was my matrix a and this was my matrix I One special property about this is that they it does commute So one times a and both of them are just going to equal a so the identity element as far as addition of Square matrices with extra zeros as the thing grows and the identity element as far as Multiplication of square matrices are concerned. We have the identity element Now this is going to be very quick and easy I want to show you the identity matrix here in the Wolfram language and Mathematica and it's simply identity matrix You see at the identity matrix and I can just Tell Mathematica what I want the size to be the in by in The grows and columns which will be exactly the same. There's one that is three. Let's do that in matrix form matrix form and So I'm using the proper notation here. Not the post-fix notation identity Identity matrix with three elements. There we go and we see very beautifully there now Let me just increase the screen size for you here. That was so tiny apologies for that So it's identity matrix with three elements and the matrix form of that identity matrix and you see You see it there on the screen. So very easy to do the identity matrix of any size Yeah on Mathematica. Let's just do one more. Let's do a nice big one identity matrix And I'm going to do it with seven so it's a seven by seven seven rows seven columns It's printed out in matrix form and there's a big seven by seven matrix matrices look the main diagonal down here all once everything else being a zero