 So again, one of the things that mathematicians like to do after we've defined something in this particular case a matrix What we want to be able to do is then to be able to do things to them And so we have the basic row and column operations So given some matrix and we'll define the row and column operations on a matrix as follows So these three basic row and column operations are n back and forth are m. That's a double arrow This just means to switch the nth and nth rows and similarly I can talk about switching the nth and nth columns So something else I can do k rn to rn that means that I'm going to do is I'm going to multiply every entry in the nth row By k and I'm going to replace the original values in that row and again similarly I can do the same thing column wise Quick note here. This is a right arrow because we read from left to right and this is fairly standard for mathematicians if you are reading a book that's geared towards computer scientists or People in other fields. They may actually use the left arrow notation and so they'll reverse this. They'll be k rn left arrow rn and It means the same thing, but the notation is not standardized And then finally we can do something a little bit more complicated k rn plus rm to rm I'm going to take the nth row Multiply it by k. I'm going to add it to the nth row and I'm going to drop all of the new values back into the nth row So my nth row has disappeared again similarly. We can do the same thing for columns And typically when we write these what we'll have is we'll have our original matrix I'll have a right arrow and then above or below or someplace close to it I'll have some description of what row or column operation I did and then my new matrix So for example say I want to apply the row operation r3 switch with r2 to my matrix M So I'm going to interchange the second and third rows So M the original matrix becomes when I switch the second and third rows Well, the second and third rows are switching but that means the first row is not going to change So I'll leave that and I'll just switch the place of my second and third row So my second row is going to be this zero zero three one and Then my third row is going to be one four negative three one Well, it's applied a different operation. So I'll apply the row operation three r2 Send it to r2 So I'm going to multiply the r2 multiply the second row everything in the second row by three And I'm going to replace the second row with what I just obtained Again, because I've done that first row and third row don't change. So I'll go ahead and keep those So only my second row is changing my first row is the same my second row is the same And I do want to multiply that second row by three and drop it into the third row So I'm going to multiply each of the entries in that second row by three. So that's going to be three Four times three is twelve negative three times three is nine one times three is three and there's my new matrix And again may be a little bit more complicated row operation minus three r2 plus r1 Get sent to r1 and I'm going to apply that to my matrix M So again r1 changes r2 and r3 stay exactly the same. So I can copy those down So I'm going to multiply the entries in row two by three I'm going to add those to row one and I'm going to replace the new values in row one And it's helpful to do this in stages minus three r2 I'm going to take my second row multiply everything in it by three. So my entries are going to be three negative three times the first entry at row two negative three times one otherwise known as negative three negative three times my second entry that's four negative three times my third entry and Negative three times my last entry. So there is there is minus three r2 I want to add the corresponding entries in row one. So that's going to be I'm going to add three zero negative one two And when I add those I get zero negative twelve eight and negative one. So here's my Values minus three r2 plus r1 and I'm going to take these and I'm going to drop these into row one So again row one changes row two and row three nothing happens to them So I'm going to keep them row two row three and then my entries zero negative one eight and negative one