 In the previous video we introduced the notion of a matrix, specifically we introduced how an augmented matrix can encode a system of linear equations. So the algorithm we're going to develop to solve systems of linear equations more efficiently involves matrices, but not all matrices are created equal, particularly from the perspective of solving systems of linear equations. The types of matrices that we're going to be most interested in are matrices in the so-called echelon form. echelon here is a French word, which basically means staircase or ladder. So an echelon form matrix, or we might just say an echelon matrix, is a matrix that we're going to build a staircase inside of, and that will be more explicit when we describe something just a moment. Also, to be more specific, these are sometimes called row echelon forms, as opposed to a column echelon matrix. As we will primarily be concerned with square matrices, that is matrices with the same number of rows as columns, the willies, the coefficient matrix is square, there's really no difference between row reduced echelon form and column reduced echelon form. So we'll put emphasis on row reducing, that is, we'll be talking about row, elementary row operations, but we'll talk about that more in another video. In this video we want to understand this definition here. So to be an echelon form, for a matrix to be an echelon form, there's three conditions that have to be satisfied. So the first condition right here is that all nonzero rows are above any rows of all zeroes. So another way of saying that is that if there's a row of zeroes in your coefficient matrix, it's got to be on the bottom. So for example, let's say we have a three by three matrix, we have something like one, two, three, four, five, six, zero, zero, zero. This matrix right here satisfies the condition I just described because there is a row of zeroes and it's at the bottom. Conversely, if we had something like one, two, three, zero, zero, zero, four, five, six, this matrix is in violation of the condition we are describing right here because there is a row of zeroes that is not on the bottom. This thing should be above that. Now with the elementary row operations, that's an easy fix. You can use the interchange operation to interchange the rows, but we'll talk about that later. If you had something like the following, one, two, three, zero, zero, zero, zero, zero, zero, that's also kosher with regard to condition one here because if you have more than one row of zeroes, which is a possibility, that's okay. They just have to be at the bottom. They have to be below all the non-zero rows. So this is a non-zero row. It's at the top. So that's perfectly fine. You also could have something like the following, one, two, three, four, five, six, seven, eight, nine. This matrix also satisfies that first condition because all of the non-zero rows are above the zero rows. This matrix actually doesn't have any zero rows, so their absence makes this statement vacuously true. It's not in violation because there's no rows of zeroes above a non-zero row. I should also mention on the flip side, if you had the following matrix, zero, zero, zero, zero, zero, and zero, zero, zero, this matrix is also in agreement with condition one because all of the zero rows are below the non-zero rows. As there are no non-zero rows, there's no row that's in violation of this rule. So this is what we mean. So all of the rows of zeroes need to be at the bottom of the matrix. That's the first part of being an echelon matrix. Condition two, each leading entry. Now a leading entry in a row of a matrix is the leftmost non-zero entry. Now if you have a row of zeroes, it doesn't have a leading entry. That's why we have to treat them separately with condition one. But if you do have a non-zero row, that means there are some numbers in the row that are non-zero. The first non-zero number is called the leading entry. For an echelon form, each leading entry of a row is in a column to the right of the leading entry of the row above it. So what that means will be something like the following. So we could take a matrix of like one, two, three, zero, four, five, zero, zero, six. This matrix doesn't have any rows of zeroes, so the first condition is automatically satisfied like we already talked about. But let's look for the leading entries in each of these rows. So looking at the first row, it's a non-zero row, the leading entry, of course, is in the one, one position. Looking at the second row, there is a zero in the first column, so the leading entry is actually in the second position, the two, two spot. And then in the third row, there are actually two zeroes there. The leading entry would be right there. So I put these little boxes around the leading entries. These leading entries are also sometimes referred to as the pivot positions. But these pivots, the term pivot makes more sense when we start talking about Gaussian elimination in the next lecture. But be aware that this idea of leading entry and pivot, these are used synonymous in this situation here. So we have these three pivot positions. And this is actually where the name echelon form comes from. That when you look at these things, you'll notice that there's sort of like the staircase of pivot positions that we're creating. That's the staircase that the word echelon is capturing here. So you'll notice that as you look at the pivots, when you go from the pivot in one row to the pivot of the next row, it moved to the right. And so this would be a matrix that satisfies conditions one and two. It actually satisfies condition three as well, but we'll get to that a little bit later here. Another example would be something like the following. Take one, two, three, zero, zero, one, zero, zero, zero, okay? This is a matrix where the leading entries would be the following. There's a leading entry in the one, one spot. There's a leading entry in the two, three position. And the third row, since it's a zero row, doesn't have any leading entries whatsoever. This matrix would also satisfy conditions one and two. It also satisfies condition three, but we haven't talked about that yet. The idea is as you move from the left to the right, the leading entries are going downwards. They're making this staircase. Now there can be a big step between rows, that's perfectly fine. But these matrices are in that they satisfy that condition. I keep on mentioning the third one without actually stating it. The third condition says that all entries in a column below a leading entry are zero. So you'll notice in these examples that if you look below the pivots, you get only zeros, okay? So those are examples of matrices which are in echelon form. Everything below the pivots is zero. The pivots make a downward right direction. And if there's any rows of zeros, they're at the bottom. So both of these matrices you see on the screen are examples of matrices in echelon form. Let's give you some non-examples for a moment. Like we said before, maybe we didn't say it. We kind of said it. Take the following matrix, one, two, three, four, five, six, seven, eight, nine, okay? This matrix satisfies the first position, the first thing. All rows of zeros are at the bottom. The leading entries are the following. Each of the three rows is non-zero and each of the leading entries is in the first position. This violates the second condition because the pivots are supposed to move rightward as you move downward. None of them move to the right and therefore this one doesn't satisfy the second condition. The third condition also is that everything below a leading entry must be zero. Now, if condition two fails, then condition three will fail automatically, okay? So we need that. We don't necessarily care about the things above the pivots yet. But that's the case. So these three conditions give us a matrix in echelon form. Let's do one more example of this thing. If we had something like the following, one, two, three, zero, two, four, and then one, zero, five. Again, just a random matrix here. The leading entries would be something like the following, one, two, one. This is not an echelon form. Notice that the leading entries don't always move to the rightward. This one needs to be more over here for that to work. There's also a problem with number three because there's a non-zero entry below that pivot. This one's perfectly fine. But this is an example of matrix that's not an echelon form, okay? So those are the type of matrices we're looking for. We'll look at some other examples in just a second. But let us also add to it a little bit stronger definition. So an echelon form is when you make that downward staircase of the pivots, of the boxes we've been drawing. There's also what we call reduced echelon form or row reduced echelon form. People will often call this as RREF for short, row reduced echelon form. If a matrix is an echelon form, it could also be a row reduced echelon form if we also have the following two conditions. Condition four says that all of the leading entries are ones as opposed to any number. And then condition five says that every number above a pivot is likewise one, okay? So if we look at a matrix like the following, one, two, three, zero, zero, four, zero, zero, zero, we saw a matrix like this before. This is a matrix in echelon form because as we look at the leading entries, the leading entries would be in the one one position and the two three position. The row of zeros is at the bottom, that's good. Everything below a pivot is zero and the pivots move to the right as you move downward. So this matrix is in echelon form but is not row reduced echelon form for two reasons. One, not all the pivots are one and second, not all the numbers above the pivots are necessarily zero. So a matrix can be an echelon form without being in row reduced echelon form. Conversely, you could get something like the following, 100, 010, 0101. This is a matrix in row reduced echelon form because the pivots are gonna be here at the ones, the one, one, two, two, and three, three positions. Those leading entries are one. There's zeros above and below them and they make the downward staircase that echelon means. But, and so this is actually the echelon form that we will see the most often. But it's not the only one, right? For example, the matrix that's above, if you tweak some of the numbers a little bit, we actually could put it into row reduced echelon form, like so. So this matrix, the pivot positions are in the same spots. I did change the numbers though. The four became, it became a one. And then also the number above where the three was now, I changed it so it's a zero. This matrix, while this one's not row reduced, this one is in row reduced echelon form. So we check the definitions, any rows of zeros if they exist are at the bottom, certified, the pivots make a downward right direction. Yep, we have a staircase. Zeros are below all of the pivots. All of the pivots are one and then everything above a pivot is likewise zero. So this is a matrix in row reduced echelon form. Let me give you two other examples of this. So we see two matrices on the screen right now, both of them are in echelon form. You'll notice these are now augmented matrix. When it comes to augmented matrix being an echelon form, you completely ignore this augmented column. When it comes to echelon form or row reduced echelon form, we only care about the coefficient matrix, the left-hand side of that line there. Now when you look at the first matrix, if we find the pivots, the first pivot would be here in the one-one spot. The second pivot is in the two-two spot. The third row has only zero, so it has no pivots. This is a matrix that is an echelon form because again, there's rows zeros at the bottom. We have a staircase of pivots and there's zeros below. That's exactly what we want for echelon form. It is not row reduced echelon form. Like we've seen before, this is not a one and there are numbers above the pivot that's not zero. Now, be aware that when you're considering a matrix in echelon form versus row reduced echelon form versus not echelon form, if there's a column that doesn't have a pivot in it, you ignore that column entirely. There's no requirement on that column because the idea about the pivots have to be one, this doesn't have a pivot, so no one has to be one. Above and below the pivots have to be zero. This column has no pivots, so there's no reference there. So be aware that this is an echelon form matrix, but it's not a row reduced echelon form. This one over here though is in fact row reduced echelon form. We have our downward staircase of ones, zeros above and below. This gives us a row reduced echelon form. And it turns out that when we're working with systems of equations, row reduced echelon form matrices is exactly what we want. And we're gonna see in just a second, why is it we like echelon form matrices and in particular row reduced echelon forms.