Welcome. This lesson what I want to do is look at a system that has no solution and see exactly how we could determine that by using these Gaussian elimination. So the system that I want to look at is the following. Is x minus 3y plus z equals to 4. Is minus x plus 2y minus 5z equals to 3. And the last equation is 5x minus 13y plus 13z equals to 8. So we'll start off the same way that we did the others and that's to look at the augmented matrix. So I look at the augmented matrix, that's going to be a 1, -1, 5 for the first column, that's the Xs. Next column is -3, 2, -13. Those are the y coefficients. And the last one will be 1, -5, 13. And then the augmented part would be 4, 3, 8. So I want to put this in reduced row echelon form and it may be that I don't have to go that far before I can determine whether or not it has a solution and we'll see as we go. So the first step, remember I do have a one in the first world first column, that will be my leading one. Below it I want to put zeros, so add roll 1, to row 2 and that would be my new role 2. So that would give me the following pay tricks. Remember the top row stays the same. But now this is a 0 when I add those two, this would be a -1, that's -3 plus 2 and this would be a -4. Right. And the last element for that row would be a 7. The bottom row stays the same 5, -13, 13 and 8. Now on the bottom row again I want a zero there, that leading one, everything in the column below it. I want to put zero, so I take a -5 multiply that times roll one, add it to row 3 and that would be my new row 3. What I get is the following, the top row stays the same 1, -3, 1 4. The middle row stays the same 0, -1, -4, 7. And the bottom row is what changes. When I perform this operation that gives me a 0 here, I have a minus 5 times a minus 3 which is a plus 15 add that to a minus 13, that gives me 2. I have a -5 times one which is a -5, add that to 13, that gives me an 8 and the last one would give me a -12. That's a minus five times four which is a minus 20 plus seven. Yeah plus eight. I'm sorry plus eight. So the next step. Remember generally what I want to do is to have a leading one there but I don't need to change that right away. What I want to do is to put it is a leading entry. So below it what I want to do is to put zeros so I can take 2 times row 2 and add that to row 3. And that will be my new row 3. And that gives me the following. Outcome. I'll have 1, -3, 1 top row stays the same. The middle row stays the same too. So what happens at the bottom rope? Well what I did is I took the middle row multiplied by two and added it to that third row. So that's going to give me a 0 here. So the first two entries are zero. What is this next entry? Well if I take two times a minus 4, I get a minus 8 and add get another zero. And then the last entry, what I get you see I actually get a 2 there. So that tells me right away that this has no solutions. Why? Well, what I have here when I look at it as an equation, it's 0x plus 0y plus 0z equals to 2. So that would tell me 0 equals to 2. And we know that this here can't happen. So therefore there are no solutions. So actually that's kind of that's actually kind of interesting. So here's one that has no solution. And generally that's what happens when you do this, you'll end up with a row that makes no sense. And that's what happens in the bottom one is that road there actually makes no sense. Thank you very much. Have a wonderful day. [MUSIC]
