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let's see if we can prove to ourselves some more reasonably interesting transpose properties so let's define some matrix C that's equal to the sum of two other matrices a and B and so any entry in C I can denote with a lowercase C I J so if I want the I throw and jth column of B C IJ and so each of its entries are going to be the sum of the corresponding columns in our matrices a and B so our i that IJ entry and C is going to be equal to the IJ entry in a plus the IJ entry and B that's our definition of matrix addition you just get the corresponding entry in the same row and column add them up and you get your your entry in the same row and column in your new matrix if the sum of the other two now let's think a little bit about the transposes of these guys right here so if a if a looks like this I will draw all of the entries it takes forever but each of its entries are I J just like that let's say that a transpose looks like this each of its entries we would call it let's see if you've got that same entry we're gonna call it a prime a prime IJ and these things aren't probably going to be the same there's some chance they are but they're probably not going to be the same but that's it's IJ entry in the I throw J's column and a transpose now the fact that this is the transpose of that means that everything that in some row and column here is going to be in that column and row over here that the rows and columns get switched so we know that we could write that a prime a prime IJ a prime IJ we're going to have the same entry that was an in that was in a J I AJ I maybe AJ I is over here AJ I is over here so this thing over here which is in the same position as this one is going to be equal to this guy over here if you switched the rows and columns I think you can accept that and you can make the same argument for B let me actually draw it out so if I break B transpose B transpose the entry in the I throw and Jade column I'll call it B Prime I J just like that just like I did for a so we could say that B Prime IJ is equal to you take the matrix B there's going to be the entry that's in the J throw and I call them these are you can almost say the definition of the transpose if I'm in the third row and second column now it's going to be what was in the second row and third column fair enough so what is so we already have what CIJ is equal to what's the transpose of CIJ going to be equal to let me write that down so C transpose let me write it over here alright C transpose is equal to I'll use the same notation the prime means that we're taking entries in the transpose so C transpose is just going to be a bunch of entries I J I'll put a little Prime there showing that that's entries in the matrix of the transpose and not in C itself and we know that C prime IJ is equal to c ji nothing new at all we've just expressed kind of the definition of the transpose for these three matrices now what is C ji equal to so let's focus on this a little bit what is c ji equal to we know that C IJ is equal to a sub IJ plus B sub IJ so if you swap them around this is going to be equal to you to swap the J's in the eyes a sub ji plus B sub J I I just used this information here you could almost view it as this assumption or this definition to go from this to this if I had an x and a y here I have an x and a y here and an x and y here I have a J and I here so the Jane and I there and a J and an i right there now what are these what are these equal to this is equal to this is equal to this guy right here is equal to we do it in the green is equal to the same entry for the transpose of a at IJ at IJ and this equal to the same entry for the transpose of B at IJ the transpose of at B at I J now what is this telling us what is this telling us it's telling us that the transpose of C which is the same thing as a plus B so it's saying that a plus B a plus B transpose a plus B transpose is the same thing as C transpose let me write that C transpose is the same thing as a plus B transpose so these are the entries and a plus B transpose right here and what is this over here what are these these are the entries right there we do the equal sign over here what are these these are the entries in a transpose plus B transpose right this is these are the entries in a transpose is the entries in B transpose if you take the sum of the two years adding up the corresponding entries so that's straightforward to show that if you take the sum of two matrices and then transpose it it's equivalent to transposing them first and then taking their sum which is a reasonably reasonably neat outcome let's do one more and I think we'll finish up all of our major transpose properties let's say that a inverse this is going to be a slightly different take on things we're still going to take the transpose so if we know that a inverse is the inverse of a that means that means that a times a inverse is equal to the identity matrix assuming that these are n by n matrices so see n dimensional identity matrix and that a inverse times a is also going to be equal to the identity matrix now let's take the transpose of both sides of this equation let me start with I'll do them both simultaneously so if you take the transpose of both sides of the equation you get a times a inverse transpose is equal to the identity matrix transpose now what's the transpose of the identity matrix let's draw it out the identity matrix looks like this you have just ones all the way down diagonal and everything else is zero everything else is zero right this is you can view this is I 1 1 I 2 2 all the way down to I n n everything else is 0 so when you take the transpose you're just swapping up the zeros right these guys don't change the diagonal does not change when you take the transpose so the transpose of the identity matrix is equal to the identity matrix is equal to the identity matrix and so we can apply that same thing here let's take the transpose for this for this statement so we know that a inverse times a transpose is equal to the identity matrix transpose which is equal to the identity matrix and then we know what happens when you take the transpose of a product it's equal to the product of the transposes in reverse order so this thing right here we can rewrite as a inverse transpose times a transpose which is going to be equal to the identity matrix if you do the same thing over here this thing is going to be equal to a transpose times a inverse transpose which is also going to be equal to the identity matrix now this is an interesting statement the fact that if I have this guy right here this guy right here times the transpose of a is equal to the identity matrix and the transpose of a times that same guy is equal to identity matrix implies this implies that a inverse transpose is the inverse of a transpose or another way of writing that is if I take a transpose and if I take its inverse that is going to be equal to this guy is going to be equal to a inverse transpose so another neat outcome dealing with transposes if you take the inverse of the transpose it's the same thing as the transpose of the inverse