Hello. Today's lesson will be on Gaussian elimination or row reduction. In this lesson, you will learn about an algorithm called row reduction or Gaussian elimination. In the previous lesson, we learned how to represent a system of linear equations as a matrix called an augmented matrix. In this lesson we will learn an algorithm that will transform a given matrix M to a matrix in reduced row echelon form that is row equivalent to M. We will explain how to use Gaussian elimination in the following example. Let's consider this example. This is a linear system, x_1 minus 2x_2 plus x_3 equals 0, x_2 minus 4x_3 equals to 4, and x_1 minus x_3 equal to 2. We first write the augmented matrix for the system. Here's the augmented matrix for the system that we have and our goal is to transform this matrix into a reduced row echelon matrix using the elementary row operation. Therefore since we're going to use the elementary row operation, these matrices that we end up, this one and the final one, will be row equivalent. Now, remember the whole idea for reduced row echelon form. I need to have a leading one, we're in luck because for this one is a leading one I want zeros below it and for each leading entry, the column it is in, I want all zeros. We are in luck since there is a one here. Our first step in what we're going to do, and this is just one possible way of achieving our goals, they're actually a lot of different ways that you can do this. But what we're going to do first is I need to put a zero here. To do that, I'll multiply the top row, Row 1, by minus 1 and add the results to the bottom row and that would be my new Row 3. If I multiply 1 by minus 1, that gives me a minus 1, add 1, I get 0, minus 2 times a minus 1 gives me 2, add that to 0 I get 2, minus 1 times 1 is a minus 1. Add that to minus 1, I get a minus 2 and a minus 1 times 0 is 0, add that to 2, and I'm left with 2. This is a new matrix that we have here, and I want to continue on here. The process is actually very simple. From here, I'm going to go down the diagonal to here at my leading one in Row 2, and what I want to do is put zeros below it. Then I'll go down here and have this be my leading one, and then I work my way back up putting zeros here, and then go up the diagonal to put a zero here. You'll see the process as we go. Our next step, since I want to put a zero below this one here, is to multiply Row 2 by minus 2 and add the results to Row 3 to get my new Row 3. So I will and multiply 1 by minus 2. First thing you should notice is that it's not going to change the zero here because I have a zero here. Multiply this by minus 2, I just get zero and I'm adding to zero. I multiply this by minus 2, add to 2 and that gives me the 0, minus 2 times a minus 2 gives me a plus 8, add that to a minus 2 gives me a plus 6, and a minus 2 times 4 gives me a minus 8, add that to 6 I get a minus 6. This is my new matrix that I'm looking for, and this is row equivalent to the augmented matrix that we started out with. My next step is I need to have, remember if it's a reduced row echelon form, and I should point out that this actually is in row echelon form, but I wanted a reduced row echelon form. What I need to do is I have a one here, remember my leading entry has to be one. To make this a one I just divide this row by 6. So I'll divide this row by 6, that becomes 1, divide this by 6 and this is minus 1, and this is a new matrix that I have. From this new matrix, now what I want to do is I want to work my way back up. I want to put zeros above this one here. The nice thing about this, if you noticed, since these are zeros here it's not going to change anything else if I multiply this by a scalar and add. The first thing I want to do to put a zero here, I multiply it by 4 and add it to Row 2. If I multiply this Row 3 by 4 and add it here, it's not going to change these two entries. When I multiply it by 4 and add, it will put a zero here in this four and make that a minus 4 and add and put a zero here and this is the new matrix that I have, and again this is row equivalent to the augmented matrix that we started off with. The next step is I want to put a zero up here. To put a zero here, I'll just multiply Row 3 by minus 1 and add to Row 1, and that'll put a zero here. Again, since these are zeros, it won't affect these entries here and I'll multiply it by minus 1 and add. That's going to minus 1 times minus 1 gives me a plus 1 and that will put a one here and successfully it put a zero here by that step, and this is the new matrix that I'm looking for. Now, there's only one thing left, because this is not in reduced row echelon form, remember for it to be in reduced row echelon form, the only nonzero entry has to be the leading one in the column that it's in. In this column here I have this leading one, but I also have a minus 2 there so I need to make that a zero. The last step in is to multiply Row 2 by 2 and add the results to Row 1. When I do that, I get zero here. Since this is zero, it's not going to change. This entry won't change any of these entries in the first row, we'll make that a zero, but this is a final matrix that I have and this is in reduced row echelon form and it is row equivalent to the original augmented matrix that we started out with. The final matrix is in reduced row echelon form and a system of linear equation that it represents is x_1 equals to 1, x_2 equals 0, and x_3 equals to a minus 1. Since the augmented matrix and the matrix in reduced row echelon form are row equivalent, the systems have the same solutions. Hence 1, 0, minus 1 is the solution to the original system. I should point out that to put a matrix in reduced row echelon form can easily be done using one of the many free online computer algebra systems such as Wolfram Alpha. The code for doing Wolfram Alpha is very simple. You just type in row reduce, open curly bracket, open curly bracket. This is the first row, 1, minus 2, 1, 0. We have to have a comma here. These brackets here actually show you what the first row is, this tells me what the second row is, and this tells me what the third row is, then I close this bracket off and you hit "Equal" and it'll give you that final matrix that we had. Thank you very much.