 So yesterday we looked at this identity element and we looked at the identity element on addition the identity element on the multiplication and now we're looking for the inverse now again You know this from school what an inverse is all I'm going to have is this For addition say my set this time is the real numbers if I have an addition three All I have to do is I it's to get its inverse I multiply it by negative one so that gives me negative three and what am I get with zero? So what I'm trying to get to is the identity element of that operation if it was Multiplication was my operation then it is a third You know three to the power negative one and that's going to give me one the identity element for this Operation so that's what we're trying to do so the same with matrices if I have a I'm looking for It's to the power negative one, but this is not really to the power negative one well Anyway, it's the way that we write the inverse and it's multiplication So we get trying to get to multiplications identity element for matrices, which is I and there's a special property here that it does commute So remember we said the matrix multiplication it doesn't commute But this is a special property that these do commute a Matrix and it's inverse so we're talking square matrices and it's not all made square matrices have inverses By the way, and we'll look at when that doesn't happen. So let's look at these two Would it be that a and a inverse give me the identity element? So let's multiply them remember this is the way that we write it One and two two and minus one and we have here a fifth and we have two fifths And we have two fifths and we have negative a fifth So the result is got to be a two by two matrix because it's two by two and two by two So one fifth plus four fifths is fit for five fifths. That's one two fifths minus two fifths loan We all the zero Two fifths minus two fifths zero guess what's coming? It's a one So that's four fifths plus minus and minus one So that's five fifths and that's a one so I get to the identity element so this this a it has an inverse and that is the inverse and You can do it the other way around do it in pen and paper if you just swap these two rounds So this was a a inverse there was a inverse try a inverse a These are gonna see you're gonna get exactly the same and that's the identity element You're gonna see you get the identity element there as well. So two questions the first obvious one is How do I get from here to there if only given this? How do I get there? That's a simple task If it's a the bigger the matrix that just it just takes longer But it's a simple task in the many classes of many textbooks. You're gonna be shown two ways of course the Mathematica It's nothing to do. This is easy and that's question number one and question number two where are we going to use this and That's a special lecture coming because That is where we get to deal with these things in a proper way And that is the proper way to look at systems of linear equations and and that is up next well First up next is going to be how to find this and then it's application in finding The solution to a set of linear equations of prop a proper way of doing that So there you go. Let's just go to Mathematica and this does have a look at this Very simple to do but very deep concepts and a concept that you have to get your you know into your head It's just gonna we've got to be part of the normal Knowledge in your head, but it's exciting stuff and it leads to Very exciting places So we here we are in the Wolfram language We see Mathematica with the Wolfram language inverse matrices. There are the two matrices that we had on the board I've already entered them there and so I'm just going to print out a in matrix form Let's have a look at that my one two to a negative one and I've just called it a I and V for a inverse and Let's do that in matrix form as well And you see there it is a fifth two-fifths and then two-fifths and negative one over five So let's do Matrix multiplication and the way that we're going to do that remember is with a period So let's do a period a inverse and let's have a look at that in matrix form the solution and Lo and behold, I find the two by two Identity matrix just to show you that we do have a commutivity here. Let's do a inverse Inverse dot a and let's print that out to matrix form as well and lo and behold also the identity the identity matrix as far as multiplication is concerned so very easy to do and It is this matrix multiplication here just to show you that this thing does exist and we didn't make a mistake on the board From here, we'll go on to the questions that I mentioned How do you calculate this inverse and then how do you use it to solve systems of linear equations?