in-exercises-1-and-2-determine-which-matrices-are-in-reduced-echelon-form-and-which-others-are-only-

Question

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

Thomas Norton · Accepted Answer

So we have a few matrices and we want to determine whether there in echelon form and if they&#x27;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&#x27;m not gonna go into what the row operations are in this video. If you can&#x27;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&#x27;ll just remind you that what the row operations are is a series of moves that you&#x27;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&#x27;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&#x27;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&#x27;re in reduced echelon form. Yes, Onley above are true and be leading zero Entry is a one and that&#x27;s in each row, and the leading non zero entry is the Onley. I don zero entry and it&#x27;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&#x27;s a form that&#x27;s even a little bit simpler. The national on for Let&#x27;s Look at what that means For a few examples, the first matrix we consider a is 1000 0100 0011 so doesn&#x27;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&#x27;s leading non zero, entry comes to the right of those above it. So for the first call for the first row, that&#x27;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&#x27;s in the second column on the road above has its first non zero entry in the first column, and it&#x27;s again the case for this last one. Okay, for the same reason, so be leading lan zero entries. Let&#x27;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&#x27;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&#x27;s see another example. Second matrix that we think about is 1010 0110 0001 Still, I&#x27;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&#x27;s ask if it&#x27;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&#x27;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&#x27;s not in reduced echelon form because these non zero entries aren&#x27;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&#x27;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&#x27;s air the ones that I&#x27;ve put a little red mark underneath. Keep that in mind. Here&#x27;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&#x27;s not at the bottom, so that means No, it is not in echelon form. And if you&#x27;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&#x27;t remember that, you really need to go back and review your row operations because that&#x27;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&#x27;s an echelon and reduced excellent form, and I&#x27;ll leave it to you. Teoh. Convince yourself that it actually isn&#x27;t reduced echelon form after doing the swap that I&#x27;ve indicated here in blue. OK, but until you&#x27;ve done that swap, it&#x27;s not national on form. And if it&#x27;s not Michelin form, it&#x27;s not in reduced echelon for so one last example, this is a bigger matrix. It&#x27;s entries or 11011 0202 to 0003 three 000 04 Okay, let&#x27;s ask the same questions. Do we have any rows that are all zeros? No, we don&#x27;t. So the first condition about having all of those rows of zeros at the bottom it&#x27;s automatically satisfied because there aren&#x27;t any rows of all zeros. The next thing that we dio is we go to each row and we&#x27;d locate its leading non zero entry. So that&#x27;s the one in the first column through the first row in the second row. It&#x27;s the two here in the third row. It&#x27;s the three here, and in the fourth row, it&#x27;s the four here. What? We&#x27;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&#x27;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&#x27;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&#x27;s also not satisfied for this matrix, because this column hasn&#x27;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&#x27;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&#x27;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&#x27;s not reduced. Okay, Thanks for watching

Arden Mccarthy · Answer

this question is asking us to determine if matrices are in row echelon form, which we can call or e f or if it&#x27;s in reduced Rochel inform just r R E f That&#x27;s in both Earth. It&#x27;s and neither. So, firstly, we&#x27;re gonna go over with each of these forms require. So for Rochelle on form, we need the leading entry to be one. And the leading entry is the first entry in the row of matrix. Second criteria is that the leading entry is in a column to the right of the previous row. Thirdly, rows of all zero are at the bottom of the magic, so we&#x27;ll draw out a general form of the matrix that is reduced echelon form. So it would look something like this. And the stars are, um, arbitrary numbers so they could be any number you want them to be. So here we can see that this meets all the requirements. The leading entries of each row are all one. Each weeding entry is in the column to the right of the previous row. We can see that with this step like pattern and rose of all zero are at the bottom of the Matrix. So overall, zero row is at the very bottom of the matrix. Okay, so now for a reduced Rochel inform that has to satisfy all the criteria of Rochel inform. But it has an additional criteria right here that the leading entry is the only non zero injury in the column that will draw that general for him here. So here we can see the difference, because all of the numbers above these leading entries have to be zero. So while here they could be any arbitrary number here, they have to be zeros so that those were the two types of forms that will be looking at. So now we&#x27;ll start and go ahead and go through some of the questions. So for these questions, what we&#x27;ll do is we&#x27;ll write them out with, um, ones further bleeding entries, zeros. And we&#x27;ll use that notation of the star to be an arbitrary number, and that&#x27;ll make it really easy for us to compare it to. He&#x27;s very interesting news that we just drew and defined, so we&#x27;ll get started on the first question. So for this one, we draw it out with one zero&#x27;s and stores we get something that looks like this, we can see that the leading entries are one. They are. They go in the kind of stacked triangle pattern, which means that they&#x27;re all to the right of each other. And lastly, our, uh, all zero is at the very bottom of the Matrix. So we know that this is definitely in or yes. However, because this is above a leading entry of one and it&#x27;s no 10 We know that it cannot be, uh, and reduced row echelon for so now, on to the 2nd 1 we&#x27;ll do the same thing here. So for this one, won&#x27;t we? We have our leading entries of one, but now Oh, sorry I didn&#x27;t put this as a store. So arbitrary number. Um, so we have our leading entries of one, but we see that underneath one of the leading entries, we have a number, and we see that this breaks this rule that, um, you&#x27;re leaving entry in the column to the right of the previous row. So the leading entries or to the right. So here we have two eating entries which can&#x27;t happen, so this is neither so onto the next one. Okay, so here we have our leading entries. Everything below them is a zero. We have a row of all zeros at the bottom of the matrix, and we see that this number is above a leading entry. So it can&#x27;t be ah, reduced Rochel inform, but it can be in our area. So for the next one, we see that again, Or leading entries are in that form that they&#x27;re in the right of each other. As we go down each row. The numbers under them are all zero entries. And again, like in, uh, see, they have this one has a number above it. So Campion, R r e f but it is in or yes. Okay. So onto the next one, just like before, we have our leading entries in the correct spot. But what support to know about this is that our row of all zeros is not at the bottom of the Matrix. So that means that this is in neither form because it breaks that criteria. However, if our matrix looked like this, we swapped these two rows with each other. It would be in our Yes. Okay. Now on to the next question. So if we look here Ah, the leading entries of each row. So the first number of each row are not staggered, so this is in neither form. If we wanted this to be in our e f, we would have to make sure that this weeding entry was to the right of the one above it, which it&#x27;s not okay and on to the last one. So here we can see that our leading entries or both ones the one in the second column is to the right of the one in the first column. The entries below this leading entry is zero, the leading entry herb. The number above the second leading entry is also a zero. So that means that this matrix is actually in reduced row echelon formed and any major extension reduced Russia and four is also in Rochel inform because it satisfies all the criteria as well. So this is actually in both reduced Rochel inform and Rochel informed

Max Sullivan · Answer

to see if this matrix is row, echelon or reduced row echelon. We look at the three conditions for being Rausch long and the final condition being produced. The first condition. Any rows of zeros are at the bottom of the matrix years. Our rope zeros, and that&#x27;s at the bottom. So we check that one off. Second condition is that each non zero has a one in the first position before, after each of the zeros. So the leading one, so to speak. So the leading one waiting one, letting once all three of our non zero rows of leading ones. So that&#x27;s that condition mint and the third condition is the leading one of vitro is to the right of the row that precedes it. And so there&#x27;s the first row, so there&#x27;s no road proceeding it second road. The one is to the right of that first row, and the 3rd 1 the row. The leading one is to the right of that 2nd 1 so that condition is met, which means that this matrix is row Eshel, Um, however, to be reduced. All the other elements, except for the leading ones, need to be zeros and we have these three non zero elements other than our leading once. And so it is not reduced. Just row echelon. So again, this one meets three conditions of the brush. Alonso, It&#x27;s Rochon, but it is not.

Martha Richards · Answer

This is our matrix that were provided and were first asked whether or not it&#x27;s in a row at Chalon form. And so to begin, our first non zero number from left to right has got to be one and it&#x27;s called her leading entry. So here Yes, indeed, this is one and it owes. And from left to right, here&#x27;s another one from left, right, We have no more leaving entries in. So we are yes, indeed in row at Chalon form because we also have to have all the zeros in the last row which you see here and the&#x27;s going and order. This next leading entry has gotta be to the right of this, the one above it. And that is the case as well. So yes, it is in a row at Chalon and it also asked for reducing row Athlon for for part B. And that means that a zero has got to be above and below each leading entry. And you can see here that that is indeed the case. So there&#x27;s zero above and below, so, yes, to reduce row excellent. And then to write out the system of equations, we have X y Z and W and our constant over here. So we have X plus three y This is zero Z and a double kind of leave a space here minus a W is equal to zero. And then the next one we have this is X Y here Z So we have one c and that&#x27;s plus a to W is equal Thio zero and then this is a zero is equal to one and then again, you can leave this out. But you also could just write out a zero is equal to zero. So these are our system of equations for this.

Suman Saurav Thakur · Answer

being a problem. Number 20 And which of Matics is given? Let us right, Matics First, this is 100 zero 3100 0000 one. Shore 11 And it was a little one&#x27;s It&#x27;ll and a Zito&#x27;s it&#x27;ll 20 first, The question is, is this in heroically inform? So, uh, living in tree in each of you is one 11 and one, but this one this one should have been to the right of this one. So no, this is not in Italy Glynn form and it must not be in. They do cycling from because if decision if this Matics is not in the road Glynn from that, it can not been to do cycling form. See documented Matics. If for a moment a matter thesis system, we questions will be written as one into x x three in the way. Why? Okay, exit not the X y, because exercise that s t u V devolution. We asked to you to get more variables. Okay, so we will be writing has one in dress as de de the equal to zero. It&#x27;s Teoh less for the it called zero tired it should be the equal Choose V plus W equal to zero the bless the blue equal to do it. Sorry. And the equal toe zero. So they should be there. System off questions We have Britain. This has coefficient off as they says de coefficient of d the SEC office interview these as co presenter v this ever coffees end up w thank you so much.

Ankit Gupta · Answer

In this caution we are given with the augmented metrics as 130 one Double zero Zito 10 four Double zero Tzeitel, Tzeitel, Zito 11 do and triple zero one, double zero. Therefore, we can observe that in first rule we have the leading element as one in second Ruby have learning Element was one in third row. We have learning element to his one before these 1st 1 off first rule all the elements below these are zero in the leading limit off second row, all element below. These are zero in the leading element off third rule The element of Bill ODIs is one. Hence it is nether Rachael inform nor it ease reduced radical inform Although the corresponding system off, the new questions can be given by X plus three y plus w equals 20 Next, we can take the the rebels as X. Why is Ed W and say s so we have explosively vipers w close to zero. Next the hell. Why? Plus four w equals +20 Next we have w plus s equals +22 and last We have w equals +20 So that is our givens is simply nearly commissions. Thank

Problem

In Exercises 1 and $2,$ determine which matrices …

Problem 1 Medium Difficulty

Answer

a.) Reduced
b.) Reduced
c.) Neither
d.) Echelon

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

Linear Algebra and Its Applications

Grace H.

Alayna H.

Caleb E.

Michael J.

Absolute Value - Example 1

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Alayna H.

Caleb E.

Michael J.

a.) Reduced
b.) Reduced
c.) Neither
d.) Echelon

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications