Welcome. In this lesson, you will learn about row echelon form, reduced row echelon form, and the augmented matrix of systems of linear equations. In a previous lecture, we learned about systems of linear equations. In this lesson, we will learn how to write a matrix representation of a system of linear equation. First of all, we've learned about the augmented matrix. Consider the following system of linear equations and variables, x_1, x_2, and x_3. The equation is x_1 plus 3x_2 minus 3, x_3 equals to 12, 2x_1 minus 2x_2 plus x_3 equals to 2. The following matrix is called the augmented matrix of the system. If you notice in the first row, you have the coefficients for the first equation and this dash line you write here is where the augmented part comes in. There you're actually adding the constant terms the 12 and the 2. The co-efficient matrix is very similar to the augmented matrix, which you don't have the 12 and the 2 there. These are the coefficients in the top row for the first equation. Here we have the top row or the coefficients for the second equations. If I consider the following system of linear equation, and what I want to do in this example is write, both the augmented and coefficient matrices for the system. Remember what we'll do is for the augmented matrix, the first row, we'll just have the coefficients from the top equation and it'll have that line and we'll add the seven to it. The co-efficient matrix are just the coefficients where the first equation here, this row is a coefficient, for the second equation, and this row is a coefficients for the third equation. The augmented matrix looks very similar to this, but only you add the constant terms which are 7, 4 and minus 10. What's the leading entry or coefficient? A non-zero row of a matrix is a row that has at least one non-zero entry. The left-most non-zero entry of the row is called the leading entry or coefficient of the row. In this matrix here M, three is the leading entry of row 1 is the first non-zero entry. Where the first non-zero entry that's furthest to the left. For the second row, two is the leading entry. The third row, five is the leading entry, and the fourth row which is this row of zeros, has no leading entry. What is row echelon form? A matrix M is in row echelon form if it satisfies all the following : The first one is all zero rows are at the bottom of the M. The second requirement is that each leading entry of a row is in a column to the right of the leading entry in the row above it. The third is all entries that are below a leading entry are zero. Now we have an example. The following matrices are in row echelon form. If you notice this is a leading entry here, here's the leading entry 2, and if you look at the leading entry that's above it, it'll be to the left of it. Same here, five is a leading entry of row 3 and two is a leading entry that's above it, but it's to the left of it. This is also in row echelon form. Notice that we have the zero row, but it's at the bottom because that was one of the requirements. All zero rows had to be at the bottom. This here's the leading entry this one of the first row and this two here is a leading entry of the second row. Again, this one is also in row echelon form. Notice one is the leading entry of the first row, and in the second row, this one is a leading entry. If you look at the row above it, the leading entry is to the left of it. These matrices are not in row echelon form. The problem with this one is that there's a zero row that is not at the bottom of the matrix. Right here you have the zero row that is not at the bottom. Of course if the zero row was at the bottom, this would be in row echelon form. This one here is not in row echelon form either and the problem that you have here is with this row 3, the leading entry of row 3 is right here. But if you look at the row above it, the leading entry is to the right of this leading entry. This one is not in row echelon form. There is another idea that's called reduced echelon form. What is reduced row echelon form? Matrix M is in reduced row echelon form if it satisfies the following: First, M has to be an echelon form. It has to satisfy the three conditions that we had before. The second one is that each leading entry is one. Some people call this a leading one. The leading entry is one, and the last one is each leading entry is the only non-zero entry in its column. Looking at these, here are three matrices that are in reduced row echelon form. Notice how it looks here. This is the leading 1. You go down to the next row, to the right of it is going to be the next leading 1 for that row. You go down to the next one, you have a leading row. Notice in the columns where each one of these leading ones are you have zeros. Again, this one is in reduced row echelon form. You have a row of zeros, it's at the bottom and again you have the leading entries are all ones and the leading entry in this row, the second row, is to the right of the leading row of the row that's above it. Same here. This one here is also in reduced row echelon form. Now these two matrices are not in reduced row echelon form. If you notice looking at this one, this one actually is in row echelon form, but it's not in reduced row echelon form because this is the leading entry here is a five and you want that to be a one. Same with here. This also would negate that this matrix from B and in reduced row echelon form the leading entry in the first row is three. Now here this case, all my leading entries are one. But this is not in reduced row echelon form, because if I look at this leading one here, the second row above it is a one. Remember one of the conditions as we wanted zero entries to be in the column that contains the leading ones. In this lesson, we learned about augmented and coefficient matrices of a linear system. We also learned about matrices in row echelon form. The next lesson we will learn an algorithm that will allow us to transform a matrix into one that is in row echelon form. Thank you very much.