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In Exercises 1 and $2,$ determine which matrices are in reduced echelon form and which others are only in echelon form.a. $\left[\begin{array}{cccc}{1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {1}\end{array}\right] \quad$ b. $\left[\begin{array}{cccc}{1} & {0} & {1} & {0} \\ {0} & {1} & {1} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right]$c. $\left[\begin{array}{cccc}{1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right] \quad \mathrm{d} \cdot\left[\begin{array}{ccccc}{1} & {1} & {0} & {1} & {1} \\ {0} & {2} & {0} & {2} & {2} \\ {0} & {0} & {0} & {3} & {3} \\ {0} & {0} & {0} & {0} & {4}\end{array}\right]$

a.) Reducedb.) Reducedc.) Neitherd.) Echelon

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 2

Row Reduction and Echelon Forms

Introduction to Matrices

Tomas J.

February 28, 2021

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So we have a few matrices and we want to determine whether there in echelon form and if they're in reduced echelon form. So what echelon and reduced echelon formed to refer to our forms that we can arrive at after row operations? A lot of matrix. Now, I'm not gonna go into what the row operations are in this video. If you can't remember, then you need to look back at that section of the text book or find another tutorial video on this site where Roe Operations Air explained. I'll just remind you that what the row operations are is a series of moves that you're allowed to make on the rows of the matrix, where you combine them with each other or you multiply them with a constant that you can use to manipulate a matrix without changing the solution. Set of the system of linear equations that that matrix represents so essentially but they are is just a tool for using maitresse sees to solve linear equations. Now, after row operations, you'll hopefully have simplified your matrix. And so what echelon form is is a simplified warm up your matrix that has two properties first property any rose of all zeros are at the bottom of the matrix. Second, the leading non zero entry. Uh, each row comes. Do the right of those above. You'll see what I mean by that. When we get into the examples where we dio, I just need to tell you what it means for the route for the echelon form to be reduced. So we're in reduced echelon form. Yes, Onley above are true and be leading zero Entry is a one and that's in each row, and the leading non zero entry is the Onley. I don zero entry and it's called him. So if those two additional conditions were satisfied, that is, If the Matrix is a national on form and the leading non zero entries for each row or a one and the leading non zero entries for each row are all alone in their column, then the Matrix is in reduced echelon form. Okay, so that's a form that's even a little bit simpler. The national on for Let's Look at what that means For a few examples, the first matrix we consider a is 1000 0100 0011 so doesn't have any rows of zeros. No, every row has at least one non zero entry. Okay, now we look at each row and we trick. Whether it's leading non zero, entry comes to the right of those above it. So for the first call for the first row, that's not really any condition, because the leading non zero entry is this one, and there are no rose above it. But in the next one, what we need to check is that the leading non zero entry, which is this one, comes to the right of any leading non zero entries from above. And that is the case because this one's in the second column on the road above has its first non zero entry in the first column, and it's again the case for this last one. Okay, for the same reason, so be leading lan zero entries. Let's say they move right as we move down. No, those two things right there tell us that this matrix is in echelon form. Then we asked the next questions. Which war are those leading entries? Equal toe ones? And they are so the leading non zero entries are ones. That's true. And we also ask, Are those leading on zero entries alone in their column? And they are beating. Non zero entries are alone in their column. And so what that tells us is sorry to go down that far that this one is actually in reduced echelon form. Let's see another example. Second matrix that we think about is 1010 0110 0001 Still, I'm gonna go a little bit quicker this time, but you can always go back. Rewind to what I said very carefully for the First Matrix. The first thing that we need to check is whether there are any rows of zeros and their art. The next thing we need to dio is going to each row and identify the leading non zero entry, which is this one for the first grow, this one for the second round and this one for the third row. And because those leading non zero entries air moving to the right as we move down second condition is satisfied. And so this is in Ashkelon. For now. Let's ask if it's in reduced echelon form. Remember what that meant was. Is the leading non zero entry equal to a one in a drill? Well, yes, yes, it is. And is it alone in its column? That is our all other entries of the column? A zero, and you can see that That's the case for this matrix. So it is in reduced echelon form, just like the 1st 1 Do something that might confuse you about this one. This column here has two ones in it. So you might be tempted to say it's not in reduced echelon form because these non zero entries aren't alone in their column. However, remember that that is on Lee a question. We need to ask for the leading non zero entry of each row. Okay, so this one was not the leading non zero entry and its roe that was this one. Similarly, this one, it was not deleting non zero entry and its roe that was this one. So having these two ones in the same column that doesn't prevent this matrix from being in reduced echelon form. Remember that all you need to ask about our the leading non zero entries. So there's air the ones that I've put a little red mark underneath. Keep that in mind. Here's the next matrix we consider. 1000 0110 0000 0001 So the first question that we ask is raise Okay, Biros at the bottom, hurting me. Well, hey, look right here. This road has all zeros in it, but it's not at the bottom, so that means No, it is not in echelon form. And if you're not initial inform, you cannot be in reduced echelon form. So it is also not and reduced aslahn form. And just to go a little bit further with this example, if you wanted to put it into echelon form, what you would need to do is one more row operation. Okay, You would want to swap these two rows. Remember that swapping rose is one of the allowable row operations. Like I said, if you can't remember that, you really need to go back and review your row operations because that's what this is really all about. If you were to do that, then you would have this row of zeros at the bottom and then you would be okay and we would go on to check whether it's an echelon and reduced excellent form, and I'll leave it to you. Teoh. Convince yourself that it actually isn't reduced echelon form after doing the swap that I've indicated here in blue. OK, but until you've done that swap, it's not national on form. And if it's not Michelin form, it's not in reduced echelon for so one last example, this is a bigger matrix. It's entries or 11011 0202 to 0003 three 000 04 Okay, let's ask the same questions. Do we have any rows that are all zeros? No, we don't. So the first condition about having all of those rows of zeros at the bottom it's automatically satisfied because there aren't any rows of all zeros. The next thing that we dio is we go to each row and we'd locate its leading non zero entry. So that's the one in the first column through the first row in the second row. It's the two here in the third row. It's the three here, and in the fourth row, it's the four here. What? We've identified the leading non zero entries for each row, and we can see that as we moved down the rows, they keep moving to the right. That was the second condition that's required to be an echelon form. So this matrix is in Ashkelon form now to be in reduced echelon form. There were two conditions. The first was that the leading non zero entries should be ones. But look here we have a two, a three and a four. So this is not in reduced s salon form. I'll just remind you that the other condition to be in reduced echelon form was that these leading entries these leading non zero entries for each rope ought to be the only non zero entry in their column. And that's also not satisfied for this matrix, because this column hasn't won. In addition to that, too, in this column has a two and a one in addition to the three. And we see that this column also has not zero entries in Okay, so it doesn't satisfy either of the two additional criteria to be in reduced special inform, but it is a national in form. If you wanted to put it in, reduce special inform you would just do a few more row operations. So remember, one of the room operations is that you can multiply rose by a constant. So, for example, in this rolled, we'd want to multiply it by 1/2 so that we could turn this to into a one and have the leading non zero and should be a one. We want to multiply this, but one by 1/3 and this one by fourth. Then we would have the first conditions satisfied. And then we would need to do more row operations to get rid of these other non zero entries in the columns. Okay, but there are other videos on this site that will show you how to do all of that. For now, The point is that this matrix is an echelon form, but it's not reduced. Okay, Thanks for watching

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