what-would-you-have-to-know-about-the-pivot-columns-in-an-augmented-matrix-in-order-to-know-that-the

Name: what would you have to know about the pivot columns in an augmented matrix in order to know that the
Uploaded: 2019-07-25
Duration: 7 min 25 s
Channel: Numerade
Description: What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?

Question

What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?

Arden Mccarthy · Accepted Answer

for this question. We want to find the values for A B and C, for which we have a consistent system. So let&#x27;s remind ourselves of what consistent needs consistent Ian&#x27;s that there exists a solution. So our first step is going to be to put the Matrix that we can create from our linear system and to reduce Echelon four. So right here we can read off the Rose and see how we got them from the equation. So the first row reads excellent. Blustery X two plus extra is equal to a second row reads negative X one minus two x two plus x three is equal to be the 3rd 1 reads three excellent plus seven x two minus X three is equal to see. So now we want to put this matrix and reduced echelon form. So let&#x27;s figure out the steps that we had to take to do so so we can see the difference between these two matrices matrix born matrix to is that this term is now zero. So we had to have done something to row two to get that answer so we can see that we said row to equal to rogue one plus room, too. And I gotta start zero there. Now we want to see the difference between Matrix to and matrix three. You can see here that another zeros knocked out. So that means that we had to have done something to row three. We had to use row one to do it. What we do is we said Road three equal to road three plus negative one. And that goddess Todd knocked out zero right here. So now we want to find the difference. Finally, between Matrix three and Matrix for So you can see that this is the big difference here. So we had to do something to our third row. Amused are three was too are too to get that answer. So now that we have our beatrix and reduce special on before I just copied our final matrix onto this cage, we can start trying to solve for values of A B and C, for which this has a concern for which this sister of equations is consistent. So let&#x27;s first see what we can find from the third row. So if we want this to be consistent, we want to make sure that this is not equal to zero. Er sorry. We want to make sure that it is equal to zero because we have our variables x one x to an x three, but we can see the zero x one plus zero extremes with zero X three has to be equal to zero for this to be consistent, otherwise being consistent. So let&#x27;s go ahead and write that down. You won&#x27;t see him on this three A was two times b plus, eh, people of zero. Well, go ahead and simplify that we get C was to be minus A is equal to zero. Now go ahead and see what information we can get from our second row. So this reads ex to was to x three equals B was a So we have that. We can see that if we solve for X two, we get X two is equal to B plus, eh? Minus two x three. It&#x27;s extra is a free variable. So we might as well see what happens when we solve for X in terms of its three variables. Lastly, let&#x27;s see what we get from this first equation. Here we get X one was three x two plus x three is equal to a So what we first want to look at is we want to go back to this equation right here. So let&#x27;s take a moment and look at that again. We have X two plus two. X three is equal to B plus, eh? So since we have a value for a up here, let&#x27;s go ahead and plug that into this a right here. So x two was two x three is equal to be waas X one plus three x two was x three. So now let&#x27;s go ahead and try and solve her beat because we won&#x27;t again to find the values of A B and C, for which the solution is consistent. So let&#x27;s go ahead and silver beast that we get x two plus two x three minus x one by this three x two minus x three is equal to be and this simplifies to negative X one minus two x two plus X three is going to be equal to be okay. So now we have our values for both A and B. Okay, so now we&#x27;re going to go ahead and try and sell for I receive value. So let&#x27;s go back. So what we have here we have C was to be minus A is equal to zero. We have our values for a and we have our valleys for beat. So let&#x27;s go ahead and start looking in those to solve for C in terms of x one x to a next three. So here we have C is equal Thio a minus to be and we&#x27;ll continue up here. We&#x27;ll put it in black and so we differentiated and we get that this is equal to negative Two tongues will be found for being which is negative exploited by his two x two plus x three. And then we&#x27;re adding that to our value up here for a so plus x one those three x two plus x three. So we don&#x27;t want to go ahead and simplify this So we have. This is equal to two x one plus four x two buttons to x three plus x one plus three x two plus x three We get that this is equal 23 x one plus seven Next to awareness X three So now we have all the values for a for B and first see so we can write those out together. And we can see that for this solution to be consistent. A has to equal x one was three x two less x three a pass to equal negative x one by his two ex too plus x three and lastly, see has to be equal to three x 47 x two minus the x three.

Arden Mccarthy · Answer

for this question were asked to determine whether or not given system is consistent. And if it is consistent, if its unique. So the first thing that we&#x27;re gonna want to do is to find some terms so inconsistent is when no solutions exist for the system consistent you that solutions do exist. And there are two kinds. Ah, there two ways that a system could be consistent. It could be consistent and unique, which means that the system only has one solution or the system can have infinitely many solutions. Okay, so we can tell that a system equations is inconsistent. If we meet, put the Matrix and reduced echelon form, there is a one in the right most column, and that will look something like this will use ass tricks to represent just random numbers. They could be anything. So this entry right here is what we mean when we say that the reduced echelon form has a one in the right first column. And the reason that this is inconsistent is because if we were to write out this fourth row as an equation, we will get zero x one plus zero x two plus zero x three is equal to one. These are these terms. Obviously all cancel out equal zero and we get zero equals one which we know is not true. So this is one of Matrix is inconsistent. When the Matrix is consistent again, we have those two possibilities. We have a unique solution, which means that all variables are leading variables in matrix for so don&#x27;t look something like this. So here we can see that all variables x one x to index three are leading variables because they&#x27;re all pivotal column. And no, hear that this does not have a one in the right. Most call that would be having one here and a zero here, which would make it inconsistent. So that&#x27;s what a matrix that has a unique solution would look like. Now, for infinitely many solutions, we need to have at least one free variable. So in a matrix that would look something like this so we can see that X one is a leading variable. Next to is the leading variable. But X three is a free variable, which means that this major X because and this system of equations has infinitely many solutions. So now we&#x27;re ready to go ahead and jump into the problem. So here is the first matrix that were given. So let&#x27;s take a look at the leading variables. So we&#x27;ll label our columns x one X to an expiry. We can see that X one is a leading variable. X two is a leading variable and x three is a leading variable. So since all variables are leading variables from her definition, we know that this means that this system is not only consistent, but it&#x27;s also unique. Okay, so now we&#x27;ll go ahead to the next one. We&#x27;ll label our columns again, exponents to an x three, and so we can take a look at this and see that this is not yet in reduced echelon form. What will we do know is because we have ones up here. We can very easily do grow operations to knock these out and make them equal to zero, right? We can do that with grow operations, and then once we do that, we can scale these leading variables to make them once again equal to one. So with ro operations, we can get something that looked like this. And since these were all, all the leading variables are all the variables air leaving variables again. By your definition, we know that this is not only consistent, but it&#x27;s also unique. Okay, now for the next one. Well, go ahead. Label or rose again for this one. Let&#x27;s do some row operations first. Look, take second row are to set it equal to our two minds are one and this gives us this new matrix. Okay, so we can notice right away that this second row is going to give us a problem because this reads zero x one with zero x two with zero x three is equal to one. This all equals zero. So we have zero equals one. And so we know that this sister of equations is inconsistent. If we wanted to go directly by our definition, we could do so of the more road reduction, and we could twinge these two rows. And so no matter what we would what role operations we would do to reduce the first and second row, we would still have, um, reduce echelon for with one in the right most called. So this is gonna be inconsistent, OK, on to our last one. So again, we&#x27;re gonna want to do some row operations. Since these two columns are exactly the same, let&#x27;s see what&#x27;ll happen if we knock one of them out. So it&#x27;s that are three equal to our three. Linus are too. What gives us this picture? OK, And from here, let&#x27;s see if we can get this too equal zero. So we should put in reduced special inform and have this baby only no. 10 injury in the column. So in order to do that will want to take our two. It&#x27;s that people are too minus or one and that&#x27;s gonna give us the following matrix. The one minus one is zero zero minus. Some arbitrary number is just another arbitrary number that holds true for this place and one minus an arbitrary number is again just another arbitrary number. And then we have this row of all zeros. So no matter what this number is, we can just divide or two bye done over. So just say let&#x27;s say our two is equal to our to debunk by store, for example, it will assume that all these air, the same number, they don&#x27;t have to be And that will give us this new matrix right here. And we can see that this is introduce special on form and for our variables. Excellent X. You next 30. You see, that X one is a leading entry and excuse the leading entry. However, X three is a free variable. So because we have at least one free variable, we know that not only is this that sort of equations consistent, but it also has infinitely many solutions.

Arden Mccarthy · Answer

for this question. We want to find the values of a for which this system equations has no solution. Exactly one solution and infinitely many solutions, So we&#x27;ll start with no solution. So first things first. We want to write out our Matrix from the given equation, so we have one. Next one plus two x two plus x three is equal to two to x one minus two x two plus three x three is equal to one and we have one x one plus two x two minus a squared minus three x three is equal to a So before we go have been started. Let&#x27;s remind ourselves of what the Matrix looks like. That has no solution. So the requirement is that the reduce special inform has a one in the right most column. So let&#x27;s do so matrix simplification. Here, our first step is gonna be extinct, are three and said that equal to our three minus are one. So if we do that first column stays the same. Second column. Cesaire Sorry, First road stays the same. Second row stays the same, and the third row become 00 And then we have negative a squared minus three minus one. And then over here we have a minus two. So what we want to dio is we want to figure out a value such that this is equal to zero, that this is not able to see her because that will give us and inconsistent system of equation. So let&#x27;s first simplify this turn over here. So we have negative a square minus three minus one equal to negative. A squared plus three on this one is equal to a negative. A squared plus two right. And we want it to be equal to zero. So let&#x27;s go ahead and do that now and solve for the A that will make this condition true. So a squared is gonna be able to to because we moved the a square to the other side. Then we take the square root but both sides and we get that a has to equal the square root of two. OK, so that&#x27;s our first requirement. Now let&#x27;s go ahead to our second requirement, which states we want a minus two to not be equal zero. So if a minus two can&#x27;t be able to zero, we want any thio not equal to two. These two requirements agree with each other, since two is not equal to the square to. So we know that for no solution to exist, we want a to be equal to the square of two. But for the next one, for exactly one solution, we know that system equations will have exactly one solution if the reduced excellent form of The Matrix has no free variables. So we&#x27;ll go ahead and we&#x27;ll take this simplified matrix here and we&#x27;ll just go ahead and put it on the other page so we don&#x27;t have to repeat our row operations. So we have 1212 two Negative, too. 31 00 negative. A squared plus two and then a minus two. Okay, so let&#x27;s continue so reduction, so we can get a closer to produce a short form. So to do that, uh, let&#x27;s first take our second row and said it equal to the first row plus the second row. And this will help us knock out this term right here. So when we do that, we get this matrix. First row stays the same second row since we&#x27;re out of it, becomes 30 for three and then the bottom row stays the same. Okay, so now let&#x27;s go ahead and do another row operation. So this row operation will help us knock out this turn right here to make it a zero. So to do that, we want to take our one and said it equal to three are one plus. Negative or two not will. Give us this Major X right here. We&#x27;ll get zero six Negative. 13 this rural states. Same. And the last room will also save them. Okay, I know it will Dio is we&#x27;ll just go ahead and swap these two roads so it&#x27;s closer to reduce social reform. So 30 for three 06 negative. 13 00 negative. A squared plus two and a minus two. Okay, so we want no free variable. So what that means is we want to make sure that this entry is not equal to zero. Okay, So in order to that, we set a squared plus two and not the equal to zero We saw for a squared. When we get that a and not be equal to the spirit of two, and if that condition holds, that&#x27;ll make sure that there is exactly one solution and this is sort of operation. Okay, So onto the last word, I don&#x27;t want to find value for a for which there are infinitely many solutions. And this means that the reduce national inform has at least one free variable so greater than or equal to 13 variables. So by the previous implication that we&#x27;ve already done a few times, I&#x27;ll just right are reflex of broad matrix. So 30 for three 06 negative. 1300 negative. A squared was two and a minus two. And we just got that matrix from right here. That&#x27;s where we got it from. Okay, So if we want at least one free variable, that means that we want toe Look at this road right here. And we want to make sure that this is equal to zero and that this is equal to zero. So in order to do that, we sent negative a squared close to you. Well, zero. And we saw for that. And we get that, um well, so yeah, we want to sell for that. And then next up, we won&#x27;t solve Bernie minus two is equal to zero. So we saw this really quickly. We know that we want eight equal to and so we&#x27;re gonna plug this, eh Into this equation will see we have negative a squared plus two equals zero That gives this negative two squared was too equal to zero. And then this gives us negative four plus two has to be equal to zero. And then we send finest and see that negative too has toe equal to zero. But we know that that&#x27;s not true. So we know that this part must be wrong and that part&#x27;s not true, which means that there are no values for an infinite solution.

Christine Anacker · Answer

All right. So it&#x27;s asking us if we have the following three. By for matrix, What would the value of K have to be in order for this system? Tau have no solution. Now, if you remember back to have no solution, that means we need to have a false statement. So we have to have an equation that doesn&#x27;t actually work. So my third row right here to make this equal into a false statement. Right now I have zero equals something. The only thing zero can actually equal zero. So this would be a true statement, so okay would have to be any value that is not zero positive or negative. So the minute I get to that, kay is not zero. That would make this a false statement. So as long as kay is not zero, this system would have new solution.

Rachel Francis · Answer

if you were to graph both equations from your system and the grass run parallel to each other, the system is said to be inconsistent. This means that there are no solutions to your system. For example, two equations. Why equals negative three X plus five and why equals negative three X minus two are inconsistent. The graphs of these equations look like this.

Ankit Gupta · Answer

solving a system of equations using mattresses. We can often come across because it&#x27;s inconsistent equations. So how do identify inconsistent equations is if we have a Mac tricks late? You won bqool in the three I&#x27;m here for B one B two B three and before C one, C two, C three and C four. Let these weak elements off a metrics which is presenting us the stomach. Three questions If we see that we have to such rules toward more satchels. But if all the corresponding elements are equal likes, everyone is equal to see one B two is equal to see. B three is equal to C three, but before is not equal toe C four. This means that doesn&#x27;t stubble of equations is in the consisting. No, it does not always have to be the last elements off. The tools will be unequal if we have any. Guests with three elements are equal to each other for the two rows, but anyone set off elements is not equal to each other. That will tell us those two equations are inconsistent

Problem

A system of linear equations with fewer equations…

Problem 28 Medium Difficulty

What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?

Answer

In order for the solution of a system to be unique each column of the coefficient
matrix must contain a pivot. The last column of the augmented matrix cannot
contain a pivot.

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

Linear Algebra and Its Applications

Anna Marie V.

Heather Z.

Kayleah T.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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In order for the solution of a system to be unique each column of the coefficientmatrix must contain a pivot. The last column of the augmented matrix cannotcontain a pivot.

Linear Algebra and Its Applications

Anna Marie V.

Heather Z.

Kayleah T.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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

Anna Marie V.

Heather Z.

Kayleah T.

Michael J.

In order for the solution of a system to be unique each column of the coefficient
matrix must contain a pivot. The last column of the augmented matrix cannot
contain a pivot.

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

Linear Algebra and Its Applications