 In the end of section 1.5, we are going to introduce a little bit of vocabulary. We're going to talk about the echelon form of a matrix. So in a matrix, if you have a row which every scalar is zero, we call that a zero row. It's kind of obvious, but we should clarify with a formal definition. Therefore, if a row is not a zero row, we call it a non-zero row. And in a matrix, if you have a non-zero row, the leading entry of the row is going to be the leftmost non-zero number in that row. Since the row is non-zero, some number in that row is non-zero, and therefore there is a leftmost non-zero entry. We call that the leading entry of that row. So with that terminology in mind, we're now ready to talk about the echelon form of a matrix. This is specifically what we call the row echelon form of a matrix. You could talk about column echelon forms, but we won't in this series. Therefore, when I say echelon form, that'll be meaning row echelon form. So we say that a matrix is an echelon form if the following three conditions happen. First, there is no zero row that's above a non-zero row. So all the zero rows are in the bottom of the system. That's the thing. So all the zeros are in the bottom. Next, each leading entry in a column is to the right of any leading entry in a row above it. And then the last one is all entries in a column below a leading entry are zero. So in other words, we're trying to construct matrices that have the following sort of prototype. We have something here, something here, something here. We have a zero, something here, something here. We have a zero, we have a zero, we have something here. This would be an example of a matrix in echelon form. So you'll notice that this matrix doesn't have any rows of zeros. So maybe we tweak it a little bit. I mean, it doesn't, if you don't have any rows of zeros, then the first condition satisfied automatically. But let's just add a row of zeros for the sake of it. The row of zeros are at the bottom. If you look at the leading entries of the matrix, there's a leading entry here, here and here. Notice the leading entries make this downward staircase. That is, if you take a leading entry, it's to the right of the entries that are above it. It makes this downward staircase. And everything that's below a leading entry is called a, well, it's just a zero, right? This would be an example of an echelon matrix. Now let me actually explain what this word echelon actually means here. It derives from a French word, which would translate as staircase or ladder. And so a matrix is an echelon form. If we're building this ladder of zeros, we can climb down the matrix along these leading entries. Like here, we take the steps down, right? These leading entries are also sometimes called pivots, a pivot position. And the terminology pivot position makes more sense when we introduce the elimination algorithm. We'll talk about in section 1.6, excuse me. We say that a matrix is in row reduced echelon form. If it's an echelon form, we sometimes call it REF for short. We say that a matrix is in row reduced echelon form. If in addition to being an echelon form, we also have the condition that every leading entry, every pivot position is a one. And every pivot entry is a one. And all entries above a leading entry are zero. So to be an echelon form, everything below a pivot must be zero. To be in row reduced echelon form, everything above and below the pivots have to be zero. And also to be an echelon, to row reduced echelon form, the leading entries have to be one. No stipulation is on that for echelon form. So every matrix which is in row reduced echelon form is in row echelon form, but the converse is not necessarily true. Like I said, when a matrix is an echelon form, those leading entries we call the pivot positions or the pivot itself. If a row that has a pivot in it, we call it pivot row, a column that has a pivot in it, we call it pivot column. Not every row or column has pivots in them necessarily. So we might have a non pivot column or a non pivot row. The number of pivots in a matrix we call the rank of the matrix. And this will be denoted as a rank of A where A is the matrix here. Let me give you some quick examples of such a thing. Our first matrix is a matrix with complex entries here. This would be, its coefficient matrix is three by three. The augmented matrix itself is three by four. This is an example of a matrix in echelon form. Let's actually check to see what's going on here. You'll notice it has a row of zeros that's on the bottom. When you have an augmented matrix and you're asking, is that matrix in echelon form or not? Ignore the augmented column. That's why we put this vertical line here. In terms of determining whether it's echelon form or not, anything past the augmented line there doesn't matter. No one cares about what's on the right of the vinculum. We only care about what's in the coefficient matrix. So it has a row of zeros, it's in the bottom. That's the first condition. Notice that the pivots make a downward staircase, right? This pivot is to the right of the pivot above it. That's fine. The fact that there's a gap here is not a problem whatsoever. That's acceptable for an echelon form. And every element below a pivot is zero. Notice in column two, it doesn't matter because there's no pivots in that column. So there's no condition violated right there. So this is a matrix in echelon form. Is it in row reduced echelon form? Well, let's double check that. To be in row reduced echelon form, the pivots have to be one, which they are. And everything above a pivot has to be zero. Uh-oh, this is at zero. So this matrix is not in row reduced echelon form. It's an echelon form, but it's not row reduced. All right. It's not already enough. On the other hand, take a look at this matrix right here. Notice there's no rows of zeros. So condition one is checked automatically. The pivots make a downward staircase. There are zeros below the pivots. So this is an echelon form. It's now an echelon form. It's in a row reduced echelon form. Each pivot is a one. So that's great. And every number above a pivot is also one. Again, the augmented column doesn't matter in this consideration here. So this matrix is in row reduced echelon form. Every RREF is an echelon matrix, but not every echelon matrix is RREF. And so that distinction, hopefully we can see here. So this gives examples of matrices which are in echelon form. Most matrices are not an echelon form. But I do want to point out to you that as we've been solving systems of equations, this matrix right here is an echelon form. Here are your pivots right here. This matrix is an echelon form. You have a downward staircase of pivots. There's a row of zeros at the bottom. Everything below a pivot is zero. It's not in row reduced echelon form because there's a nonzero entry above the pivot, although the pivots are one, which is great, but not required. So this matrix was in echelon form. And one of the keys about solving this system of equations, basically if we summarize things as we want to convert using row operations, we want to convert an augmented matrix into echelon form, and then we can solve that matrix using back substitution. That in a nutshell is what one means by Gaussian elimination, which we'll introduce formally in the next section, 1.6.