5

Rank and Nullity e Matrix. The folkwing nmtrices md B IATu eqjuivaknt:md B =() [2pts| Find the rik o AJ Explin sour #nsWOT ,[2pts | Find the nullity of A, Explain ...

Question

Rank and Nullity e Matrix. The folkwing nmtrices md B IATu eqjuivaknt:md B =() [2pts| Find the rik o AJ Explin sour #nsWOT ,[2pts | Find the nullity of A, Explain sour answOT ,[Spts | Find Iwusis for the rOw" ~puce 0 A Explain sour nnswtr .(d) [Spts | Find Iuusis for the colun ~pMt of A, Explain Tulu Muta

Rank and Nullity e Matrix. The folkwing nmtrices md B IATu eqjuivaknt: md B = () [2pts| Find the rik o AJ Explin sour #nsWOT , [2pts | Find the nullity of A, Explain sour answOT , [Spts | Find Iwusis for the rOw" ~puce 0 A Explain sour nnswtr . (d) [Spts | Find Iuusis for the colun ~pMt of A, Explain Tulu Muta



Answers

Find the rank and nullity of the matrix; then verify that the values obtained satisfy Formula 4 in the Dimension Theorem. (a) $A=\left[\begin{array}{rrr}1 & -1 & 3 \\ 5 & -4 & -4 \\ 7 & -6 & 2\end{array}\right]$ (b) $A=\left[\begin{array}{rrr}2 & 0 & -1 \\ 4 & 0 & -2 \\ 0 & 0 & 0\end{array}\right]$ (c) $A=\left[\begin{array}{rrrr}1 & 4 & 5 & 2 \\ 2 & 1 & 3 & 0 \\ -1 & 3 & 2 & 2\end{array}\right]$ (d) $A=\left[\begin{array}{rrrrr}1 & 4 & 5 & 6 & 9 \\ 3 & -2 & 1 & 4 & -1 \\ -1 & 0 & -1 & -2 & -1 \\ 2 & 3 & 5 & 7 & 8\end{array}\right]$ (e) $A=\left[\begin{array}{rrrrr}1 & -3 & 2 & 2 & 1 \\ 0 & 3 & 6 & 0 & -3 \\ 2 & -3 & -2 & 4 & 4 \\ 3 & -6 & 0 & 6 & 5 \\ -2 & 9 & 2 & -4 & -5\end{array}\right]$

Hello there. So for the following exercise, we need thio way Have some matrices and we need thio Calculate the rank off the matrix on the nullity off the metrics on. Well, that's that. The information off these two, basically we got some metrics. And what the serum say is that Suppose that a is in our three? So is a matrix off sm matrix three by three on the reels. Okay, so basically the the serum say that the rank off a plus deniability off the metrics, A should be close to three Onda. Uh, well, we need to to test that. Or which is the same that the rank off the metrics a plus, the nullity off the metrics. A should be close to the number off columns. Uh, off the metrics. A Okay, so well, we're going to put, uh, this on the test. So we're going Going to just basic. So let's start. Okay. So for the first metrics, um, we got the following one minus 13 five minus four minus 476 on to Okay, So this is our matrix. So how to compute the rank off the matrix? Basically what we need to do is to reduce the Matrix to the Asian form by performing some role operations. I'm going to do the full procedure for this the first metrics. But for the following, I am just going to put their reduced form. Because otherwise this video will be really, really long. Okay, so with this exercise for this, I'm going to show the whole procedure. So first, basically, the Asian form means that we need to put zero here. You're here. You're here on put once here. And here's how to do that. Well, we just need Thio. Make some raw operations on. You will see. That is not that complicated. So I the first step, I'm going to put zeros on this part on this part here on seven. On the five I try to I will try to put a zero. Okay, so the second row I'm going to put the operation here will become will be the second row minus five times the first roll on for the third role. What I'm going to do is multiplying. Well, basically, will be our three. Here is our story. So here will be Are the third row minus seven times the first road. With that, you will see that we're going to paint a one on zero A zero here and here. Okay, So the film role remained the same on the second one Started changed 01 minus 19 and hear the same. Okay, so here you can notice something. So these two roles are the same. So basically, in the next step, I'm going to put zero on this whole row. How to do that? Well, we just need to subtract. So the operation is that are is there are three thorough will become the thorough minus the second row. Technically, so here we have taken 01 minus 19. And here's 000 So this is the Asian form, the reduce it nation form of the Matrix. Here, you can see that we need for the rank. Okay, So the rank off this matrix a will be the number off rose on the reduced action form of Dimitri's. A So, Or the number off the Yes. He is just a number off overalls that we got here. The number off Bibles, Basically, that's the real definition. So here the rank is just because we only have two pilots here, here and here. So the rank off these metrics is equals to do just by using the theory, Um, we can attain the nullity off the system. However, we way. Actually, that's what we want to show that the, uh the theory is true. So we need to make the whole compilation. So how to compute the knowledge? So let's go to the community or other called other people. Call it the colonel off the metrics. So basically, we got we need to solve the system. A X equals to the zero vector. So our matrix, um, you're going Thio considered to reduce it form means these metrics here. So, uh, here we got that one more step, actually. So our vector is one minus 1301 minus 19 000 We can put a zero on this place here. So how to do that? Well, we need just to subtract this deep the heroes. So the first role will become the first Rob lost the second row. So this is 10 minus 16, 01 minus 19 00 Just to make it easier to calculations. So there's going to be or matrix a on we need to a dimension before us off this system. But this is equivalent to help the following So x one minus 16 3 on X two minus 19 x three is equals to zero on here is equal to zero. So we need to find some nontrivial solution on uh, well, for the nontrivial solution, we just we have some free variable that in this case we can 233 x x three is a is a free variable s o x three. I'm going to change it as t I'm going to represent it as just another constant, some constant t. Okay, So by doing that, the solution becomes x one. Is it close to 16? T on X two is equals to 1980. So the vector a solution for the system is given by the following its 16 de 1990. And we say that ex Christie So this corresponds to our X that the solution for this system here on this is the the The solution on as you can see, is just one vector. You cannot separate this. So the solution X is just We can represent it in this way. 16. 19 1 on. He's just made by one vector. Okay, so is any scale. Basically, you can put anybody here. So the newly space, the newly space eyes, a span by only one vector that is 16, 19 and one. Okay, so the whole null space it generated by scaling scaling vectors off this form. So it's just much buying by any tea. Here. This'll vector on the obtained, the solution for the north space. So the new space eyes generated by only one vector. So the Newell nullity off the matrix A is equals to one. Yeah, Onda, we got all that we need for to prove the the theory. Um, the chick, the theory, Um so the rank off a plus, the nullity off the matrix a should be equals 23 And that is true because the rank is a constitute on the nullity is equals to one. So that is three. So that's correct. Okay, so this is the basic procedure Thio off in the role of the rank on the ability off the vector. So for the next, I'm not going to make the whole procedure to reducing the matrix because could be too long for it could take us too long. Okay, so let's continue. The next matrix that we got is the following 20 minus one. Here is for zero minus 2000 Okay, so here we got some, uh, we see that We've got something here. Okay? So, actually, you can observe that these matrix, this role here is to times these the first role. So the operation that we need to do is that the second role will be equal. We're going to make an operation here, so we'll be, um, two times the first role, minus the second role. And by doing this, I'm going to put a zero on this place. But actually, what I'm going to do is put in the zero to the whole row. So at the end, we only have these matrix so clearly here we only have one role under on the Asian form of the Matrix. So the rank is just one. So the rank off a is equal to one. No. Now, let's continue to the nullity. Okay? So we need to technically just solve the system. A X equals to the zero vector A will be the reduced form. So it's 20 minus 10000 times x one x two x three equals to the zero vector. Okay on. Well, the solution is given in terms. Will is just X one is equals to one half off. Three. Okay on. Well, x two, it doesn't could be any value. Okay? Yeah, ex. You could be any value. I'm going to put it equals to t. And actually x three am going to put it equals to it because it is also free variable. So here we got the solution for the new space for these systems solution for the system. So the solution is given in this way. So is the vector. Ah, that is one half s t s. Okay, so look, we can separate this vector here. So is equals to t 010 plus s one half 01 Okay, So what does mean is that the new space, the new space off the matrix A is a span by the vectors. 010 Um, one half 01 The whole most space is generated by linear combinations off these two vectors. That's what the meaning off this that this T and s could be any value could be any in the real value is that here? We got a linear combination off two vectors, and these two vectors generate the whole new space. So the nobility is just the number off the generators off in the space. So the nullity off the metrics A is equals to two because we got two vectors here. Or you can see it as we got to Victor's here as a linear combination. This here. We cannot make this DISIP oration on the previous example because you can see that t is multiplying the whole vector. We cannot separate. Okay, so the in this case, the spaces just generated by only one factor. However, here you can see that is generated as two vectors because you need the linear combination off to off these factors. So in this case, the reality is equals to two. So we got that the rank off the matrix is equals to one. The nullity is equals to to, so we can check that the rank off a plus, the nullity off a is equals to three. Because this disease is, uh, yeah, this rank here is equals to one. This note it is constituted is equal to three. So they're equal. So we have checked that this you satisfied for these metrics? Let's continue. Let's see the next matrix. So the next matrix is 1452 2130 minus 13 to 2. Okay, In this case, note that the column vector the number of columns is four. So, basically, in this case, we need to check that the rank off this matrix plus the nullity off the matrix should be equals to four. Because in this case, these vectors are on our four. Okay, so we need to satisfy duck. The period examples is a matrix that have three columns. So we need to check this case are four. So the theory and say that we need to check that the round personality should be equal to four. Okay, so let's continue. So, after reducing these metrics to the Asian reform, I'm not going to put the procedure, protect too much time. So after making some row operations here, we obtain the following Asian reduced form off the matrix here. Zero want want here for divided seven? I hear 0000 Okay, so here the rank is equals. Due to we got only two roles on the reducing the Asian for so the rank off the metrics a easy constitute. That's the first thing. The second part is competing minority. So, as I mentioned before, and as we have done before, we need to find a solution for this system. The Matrix A is Chris correspond to this one. So what we've got here is 101 minus two. Divide seven 0114 The way. The seven on dhere 0000 that multiply by x one x two x three x four and this should be equal to 0000 Okay, so the trade find solution for the system. So here for so X one is equal to so the first. Okay, so here we got three equations on forum knowns. Actually, we got only two equations here. Two equations on the forum known. So we're going to obtain to free variables. Okay, Those free variables will corresponds to x three and x four because it doesn't have any pilot on that on the metro. Okay, so x three and x four are going to be your free variables. You can You're going to to observe the for example, in the X one. The X one is, of course, to minus 63 plus two divided seven x four on the solution for X two is equal to minus 63 minus four divided seven x four. Okay, so the solution is given in terms off X three and x four, so we could put any value to them. Also, X three could be s on X four is going to be t. So after doing this, the solution The solution for the system in this case X is equals to minus s plus two divided 70 minus s minus four, divided 70. And here s anti. Yeah, so you may observe that here we can separate this. So we on one side we had we got s that is equals to minus one minus 110 on T on the other side. So here we got two divided seven here, minus four, divided 70 and one. So, in this case, the newly space is generated by linear combinations off these two vectors, as I expressed before, basically What I'm saying is that the newly space off a is a span by linear combinations of these two vectors, minus one minus 110 on the vector two divided seven minus four. Divided 701 So the nullity off this vector, so at the end, nullity off the victory. A easy constitute on us. Uh, as as we expected, the rank off a plus, the nullity off A This is equals to two plus two. This is equals to four. And we expected to have tea before. Okay, because it should be equal to the number of columns. The number of columns off the matrix A on that. Okay, so that is, uh, true. We have check. Let's continue. Yeah. So the next matrix is the following one for 569 three minus two one for minus one. My number 10 minus one minus two minus one, 235 78 Okay, so this is the next matrix in this case. Note that the number of columns is equal to five. So in this case, we need to check that the rank plus the nullity should be equals to five. So this is what we need to check. So just remember that the number of columns is equal to five. Okay, so here again, we need to reduce the Matrix to the Asian form. So after performing some raw operations, we obtain the following matrix 10 1 to 1 01111 0000000 Okay, so, after reducing the metrics to the actual informed within the following on Dhere, you can observe that we got only two rows. So in this case, the rank off the matrix is equals, Tutut. Now we need to check The nullity toe again is a X equals to the zero vector in this case is equivalent to say, 1012101111 00000 the mhm, X one x two x three x four X five equals 20000 Okay. Okay. So look. Uh huh. Here's the one. Okay, so here we got only two equations. Three or known? 50 no. So, technically, three are going to be free variables, and we're going to check that so x one is equal to minus x three minus x four two times x four on minus x five The x two is up is equal to minus X 63 minus x four R minus X five and that's all. So that is the solution. So here, x three exploring an X five are free variables we can put. Actually, we need to put some constants there. So let's, uh, say that three Ys equals two s. X four is equals to t. The next five is equals to are so could be any value. Could it could be any in real value here. So if we do this, replacing this to the solution will think that X will be, um, here minus s. So here is minus s minus two team my news are minus s minus D minus. Are S D r. Okay, on you can see that in this case, we can separate this dissolution in is made by a linear combination of three factors. So what we obtain is the following s minus one minus 1100 plus T minus two minus 1010 Um, and those that are most by by our so here will obtain minus one. So here is minus one my 1001 So, in this case, the new space the responsible solutions off this system are represented at linear combinations off three vectors. 123 formerly has expressed it on the other example. On the on the other exercise, it means that the new space is a span by these vectors minus one minus 1100 plus minus two minus +1010 on plus minus one minus +1001 Okay, so it it is a span. Here is a separate commissary by cons. Okay, so here the new spaces generated by three vectors. So the new space, the reality actually, the reality off The matrix A is equal 23 Okay. And here previously, check the Iraqis equals to to so v rank off a plus. Nullity off. A should be equal to the number off columns. But the number of columns is equals to five. And here the nudity is equal to three on the wrong Waas equals Tutut. So this is true Good. It was very fine. Finally we got the last metrics. Okay, so the last metric that we need Thio do this whole procedure is a following. So here is one minus 32 to 1 036 zero minus three. Tu minus three minus 244 396065 minus two nine to minus four on minus five. So now we've got these metrics. Okay, so in this case, again, we got 35 columns. So the number of columns with this So the number of columns is equals to five. Okay. So, again, we need here to reduce these metrics to the action form. Here is for just in case. Okay. Okay. So after performing some row operations, we obtained the reduced form off. They reduce it matrix to the veteran form on it. Corresponds to do for thes matrix 100 t seven divided six 01 00 minus one. Divine six 001 zero minus five by the 12 on Dhere to rose just off zeros. Okay. Yeah, So here we just need to count the number off Rose here. We got three rows. So the rank off the metrics a easy close to three. The number off rose off the reduce natural. Now, let's compute venality, so we need to find the solutions off the system on that corresponds to 100 276 0100 minus one Divided six 0010 minus five Divine 12 00000 And here 0000 on this multiplied by x one x two x three x four x five, and this should be equal to 00000 Okay, so there's the system that we need to find a solution. So x one is equal to mhm minus two x four minus seven divided six x five x two is equal to 16 x five and finally X three is equal to five divided 12 x five. Okay, so in this case, the free variables our X four and X five So x four I'm going to write equals two s on X five is going to be equal city to the solution in this case, for the space is X. Here is minus two s minus seven divided six deep one six de five divided 12 de s anti. Okay, so in this case, we can separate this system into just two, so those that are multiplied by s or here is minus two 0010 On those multiply lights to hear you. The minus 7/6 one divided six five divided 12, 0 and one. So in this case, the new space is generated by two vectors. So as we have seen before, then the little will be just too. So the knowledge team off the metrics, A will be equals to do so the rank. So we need to verify that the rank off a toe, the nudity off a is equals to the number off columns off the metric A. Yeah, the number of columns off the matrix A is equals to five. The nobility is equals to two. And we check that the rank WAAS equals 23 So this is true on that for this exercise.

And this exercise we're going to use to facts. The first fact is that if a sin matrix of size M times in, then the nullity off a plus the rank of a is equal t o n the number of columns on second that for any matrix, the rank of A is the same as the rank of Adrian's posed so in exercise. Still, the Matrix given is of size three times four. So we have that the finality of a, um, equals for my news. The rank off. Hey, this is why you see, in fact, a on board by using fact be, well, this is four minus the rank of a transposed. And then if a is a matrix of size three times four Well, we have that a transposed is a matrix of size four times three. It has three columns. Then during the rank off, a transpose is three minus the nobility of a transposed on. From this expression, we have tamed the following. Now the T f a ace one pleasant ality off a transposed on This is Theis expression that we were asked for in part. Hey, now for part B, the President is the same. Well, similar. If is a matrix of size M times n then the reality of a is and the number of columns minus the rank off a onda We know that the rank of a is the rank of a transposed. So we have this. Um Well, now remember that a transposed in this case is going to be a matrix of size and times M yeah. On dso we have the rank of a transposed is in this case, m minus the ability of a transposed. From here, we can clear that finality off A It is the same us and minus, um, minus class. The nullity off aprons post on This is the expression for part B.

Okay, So what we have here is a prepaid for May thinks we're in. We're asked to get its reduced role as you import minutes round. So based from our previous lessons, we need to convert this first with its role. Excellent form. So that means it's drunk and to go further operations to convert it to reduce grow. Actually, for so the industrial accident for has the same operations Withrow excellent form. But that's different. Very here. Yeah. So we say that a matrix is in its reduced erosion and port if it satisfies the requirements first, the first non zero number in the first of all is the number. What second? The second row also starts with the number one, which is further to the right than the leading into the first room for every subsequent room, the number one must be further to the right, so it should look like this. We sure Sambo sister keys they're living and in each row must be the only non zero number it's called, and finally, and in an zeros are placed at the bottom of the matrix. So we can do this by performing syriza elementary row operations, including inter changing wonder with another multiplying one grow bananas zero constant and replacing one drug with one drop plus a constant times another grow. So with this example, we're going to convict this first its role petulant for okay, So first step A What we're going to do now is the trans seldom leaving for efficient off artery by doing all right, buddy is equal toe artery minus 2/3 off are you by doing that? We now have zero one go one zero ready one who and 00 negative to over three and negative 1/3. If you are, don't notice this is already in its role echelon farm. So we are asked to the meat It's Trump and by definition, round is the maximum number off leaner, independent foreign Imitrex, which is equivalent to the number off non zeros in its role, petulant the tricks. So for our example, we have 12 and three rows with non zero element which is rolled. One broke through in the three, therefore our runk history. So we're not done yet since we are asked to get its reduce grow at you, look for so for over next step We want to make zeros in calling toe. Except the entry. A throw one called to good. This we need to sub drop. But he are one from art. Now we have zero one don't. One 00 negative. Five negative one and then 00 Negatives to over three and negative. 1/3 for a step. See, we want to make zeros in column three except the entry A troll toe fall of three. To do this, we need toe divide Are too. We'll make it the fight by doing that, we now have zero one. Oh, one 001 one over. Fine. 00 Negatives to over three and negative 1/3. We're a next step. We are going to send it up. So I do. From what? That we now have zero 10 30 over. Five 001 1/5 and 00 Negative over three. And they get the one over the next. We want to add. Roto won't be played by 2/3 to wrote three. So by doing that, we know how. Zero 10 pretty over five 001 in 1/5. Then 000 negative one Oversight for step if we want the MiG zeros in column four Except the entry a throw pre called before to do these We had raw three multiplied by 301 By doing that, we now have 0100 zero 01 in 1/5, then 000 negative one over. Fight for step G. We want to I proto and what city. And with that we now have cereal. 100 0010 000 Negative 1/5. So far, our find us that we are just going to multiply artery win negative fight. So our reduced Groeschel inform therefore is 01000010 and then 000 one. So this is our final answer with cheese The reduce grow excellent for off the meetings.

Jean. We're getting matrices and rest defying all the ideas, values and corresponding linearly independent Eigen vectors from these matrices sneaking into his house and were also asked to impossible to find a non singular matrix P. That diagonal eyes is the matrix I wrote the Staten Island Ferry the other night and went into and somebody had just left in part A. Were given the matrix A. This is the two x 2. Matrix 2. -3 2 -5. Show your hair. Now this is a two x two matrix. So we can find the characteristic polynomial delta T. This is T squared minus the trace of tea which is negative three. So T squared plus three. T. My plus the determinant of A. Which is negative four. If we can factor as T minus sorry, T plus four times t minus one. And so the matrix A. Has Eigen values lambda one equals -4 and landed two equals 1. Yeah. Absolutely hard fuck. Unless quickbooks starts making and diet, energy drink. I think nobody listening to this show if there's if there's a way to harass quickbooks payments yes for once do something controlling house. Yeah. Now to find the investors corresponding police Eigen values, it's a quick it's all, it's all company called into it into it. Lobbies the government to keep taxes complicated so they can continue robbing. Oh boy. So in the case uh lambda equals negative four. We're going to subtract a negative forward down the diagonal of A. And so we get matrix M. Which is a plus for I Which gives us the two x 2. Matrix 6, -3 2 -1. This corresponds to the homogeneous system. Six X minus three. Y equals zero and two X minus Y equals zero. Which just corresponds to the single equation two x minus Y equals zero. Distillation of professors trying to be. So let's take for example X equals one. Then it follows that Y equals two. So such an Eigen value could be the equals 12 And this is an Eigen vector belonging to I can value negative four. Yeah. How do I get the cheapest? A lot of things have changed since that time. So yeah, you could, you could say now consider the other. I can value Okay, Landed two equals 1. We're going to subtract one down the diagnosed today to get the matrix M so ends a minus one eye or a minus I. This is the two x 2. Matrix 1 -3 to -6. So which corresponds to the modernist system? Let's see X -3. y equals zero and two X minus six. Y equals zero. Which corresponds to a single equation, X minus three, Y equals zero. Listen, find an Eigen vector, We have one degree of freedom. Take white to one and then X. Is three. So we get the vector U. Which is 31 And this is an Eigen vector was it? Yeah. Makes a lot more so belonging to the Eigen value linda. To which is one. Why breathe believe me. Trust the covers. Okay. It was now in part B. Was it? Yes. What is indian jewish department? Yes there there are. It was unintentional but in part B. Yeah. Were given the Matrix B which is the two x 2. Matrix 2 4 -1. 6 should do that more often. This is also two x 2. Matrix. So to find the characteristic polynomial delta T. Well it's T squared minus the trace of B which is eight times T. Plus the determinant of B which is 16 And this is equal to T -4 square. Mhm. Therefore the zeros of our characteristic polynomial which is just landed equals four. RR Eigen values. We acknowledge it appears after you took a ship. Matthew. Such a big ass. Mm hmm. I got to say really Now for the siding value to find corresponding Eigen vector We'll subtract four down the diagonal of a b. So we get them which is please A -4 I which is the matrix, you know -24 -1, two. And this corresponds to the homogeneous system negative two X plus four. Y equals zero and negative X plus two. Y equals zero. Which simply corresponds to the single equation negative X plus two, Y equals zero. You plug in. Y equals as they are with lines. Yeah servants and like well like just discontinued service even in boston. Keep right one second half the time you're like, well just it's not right. Come for. Yeah. Well if we take Y equals one, I just Then we get x equals two. And so we have the icon vector V. Which is 2 1. And this is an Eigen vector yeah, belonging to the Eigen value. Lambda equals four. There's only one million nearly independent icon vector park slope. Somewhere cute. For sure. All right then in part C forgiven the matrix C. This is the two x 2. Matrix 1, -4 3 -7. Her is now why? Because I like one story house. You do. I like a big story once again this is two by two. So the characteristic polynomial delta is T squared minus the trees 96 plus 60 plus the determinant plus five. Which we can factor as uh T plus two times T plus three. And the zeros of this characteristic polynomial are Eigen values so that we have item values And the 1 -2 and blame the two which is negative three. Yes, it's you. Yeah, that's right. Consider the first again. Value. Go ahead One lesson this room which you wanna fucking unless you see it. Shit Houston. I'm sorry I made a steak here, so suck my balls. But I don't like in your mouth man. Yeah. Show me stuff when you see something, teach me how to. Yeah. Sure. Hey, I'm not even playing with your homes. You want to take me more basement? Maybe you he's out fucking dress up. Sure the ship. This should be T. Plus one times T. Plus five. So the Eigen values are actually negative one and negative five. Okay so now we're dealing with the Eigen value negative one. We're going to subtract negative one down the diagonal of C. So we add one. We get the matrix M. Which is C. Plus I. Which is the Matrix 2 -4. Three negative six. As southern California which corresponds to the system two x minus four. Y equals zero and three X minus six. Y equals zero, which corresponds to the single equation x minus two, Y equals zero. And if we plug in Y equals one to get X equals two. And so a possible Eigen vector is you which is 21 possible. And this is an Eigen vector, that's it belonging to the Eigen value. I am the one which was negative one email list. Now we do the same thing with the other Eigen value in 85. So I subtract negative five down the diagonal of C. We get the matrix M which is C plus five I. And this is the matrix six negative four. Three negative two password. The reason this corresponds to the system six x minus four. Y equals zero and three x minus two. Y equals zero. This corresponds to the single equation. Three x minus two, Y equals zero. If you plug in, y equals three then it follows the excess too. And so to get the vector V which is 23 And this is an Eigen vector yeah, belonging to The Eigen value landed two which was -5. You see you're not caesar. Now on top of all of this, Looking back at our matrices, A three C. Which ones are diagonal Izabal. Well, we see that because A and C. Have a full set of linearly independent I conductors. The beat is not only A and C. Our diagonal Izabal The call that be only had one. We're nearly independent. Eigen vector in the bathroom. Right? All right. And in order to diagonal eyes A and C. Well, for A P is going to be the matrix whose columns are the Eigen vectors U. And V. So we have You which was 31. Yeah. And the second column V. Which was 12 Watch me get nasty. I don't care. And you for the matrix C. Will take P. To be the matrix with columns you which is 21 And v. Which is 2, 3. Yeah. Which is the perfect package to blow more through.


Similar Solved Questions

5 answers
Long solenoid that has 7010 turns uniformly distributed over ength of 385 produces magnetic field of magnitude atits center. What current required in the windings for that to occur?Need Help?atManieni
long solenoid that has 7010 turns uniformly distributed over ength of 385 produces magnetic field of magnitude atits center. What current required in the windings for that to occur? Need Help? at Manieni...
5 answers
10.0 Q 25.0 v N 10.0 $ M5.0! MJn 10920.0 #6I MFind the current in the 20.0-Q resistor and the potertial difference between polnts ana (Vbshown in the figure?Yanitiniz:4) 1-227m4 Vb-Va-5.63 VB) (-327 m4 Vb-Va-6.68 C) |c27mA, Vo-Va-6.68 D) (-427mA Vb-Va-4,68vE) 1-327m4 Vo-Va-5.68V
10.0 Q 25.0 v N 10.0 $ M 5.0! M Jn 109 20.0 # 6I M Find the current in the 20.0-Q resistor and the potertial difference between polnts ana (Vb shown in the figure? Yanitiniz: 4) 1-227m4 Vb-Va-5.63 V B) (-327 m4 Vb-Va-6.68 C) |c27mA, Vo-Va-6.68 D) (-427mA Vb-Va-4,68v E) 1-327m4 Vo-Va-5.68V...
2 answers
Propose reasonable synthetic route for the following reaction:CH,CH C-CZn(Hg) HCI(11?(21?AICl;COOHHNO,(41?KISO4
Propose reasonable synthetic route for the following reaction: CH,CH C-C Zn(Hg) HCI (11? (21? AICl; COOH HNO, (41? KISO4...
5 answers
40. Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.V+5"
40. Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. V+5"...
5 answers
Dx =? 1 Vx? +4
dx =? 1 Vx? +4...
1 answers
You are given a definite integral $\int_{a}^{b} f(x) d x$. Make a sketch of fon $[a, b] .$ Then use the geometric interpretation of the integral to evaluate it. $$ \int_{-2}^{3}(2 x+1) d x $$
You are given a definite integral $\int_{a}^{b} f(x) d x$. Make a sketch of fon $[a, b] .$ Then use the geometric interpretation of the integral to evaluate it. $$ \int_{-2}^{3}(2 x+1) d x $$...
5 answers
Find the arithmetic mean for each set of values.$$5,9,8,2,4,7,3,2,0$$
Find the arithmetic mean for each set of values. $$ 5,9,8,2,4,7,3,2,0 $$...
5 answers
21. (6) Identify the relationship between these pairs of compounds (constitutional isomers, enantiomers, diastereromers, or identical compounds)OH Ntbc;s4c0n5 NHzHoOHHO
21. (6) Identify the relationship between these pairs of compounds (constitutional isomers, enantiomers, diastereromers, or identical compounds) OH Ntbc;s 4c0n5 NHz Ho OH HO...
5 answers
Questlon Completion Status:Tne line . acllon of Ihe 3000-Ib force runs through ine pointsa5 shown the fgure. Determine Ine x and y scalar components 0l F;B (8, 6)3000 IbA(-7,-212650 @nu Fy 21412 Ib-2050 -1412in32650and Fy 70500uearinVoiaato tne 024t queatlon Drovents Cltner
Questlon Completion Status: Tne line . acllon of Ihe 3000-Ib force runs through ine points a5 shown the fgure. Determine Ine x and y scalar components 0l F; B (8, 6) 3000 Ib A(-7,-21 2650 @nu Fy 21412 Ib -2050 - 1412in 32650 and Fy 7050 0uearin Voiaato tne 024t queatlon Drovents Cltner...
5 answers
If a line has $y$ -intercept $(0,4),$ then 4 can be substituted for $_______$ in the equation y=m x+b
If a line has $y$ -intercept $(0,4),$ then 4 can be substituted for $_______$ in the equation y=m x+b...
4 answers
A solenoid has 2000 turns, 20cm length, and 1.0mm radius. When acurrent flows through the solenoid, it stores 0.20mJ of magneticenergy. What’s the current circulating in the solenoid?a. 22 mAb. We can't solve without knowing the inductance of thesolenoid.c. 0.22 Ad. 2.2A
A solenoid has 2000 turns, 20cm length, and 1.0mm radius. When a current flows through the solenoid, it stores 0.20mJ of magnetic energy. What’s the current circulating in the solenoid? a. 22 mA b. We can't solve without knowing the inductance of the solenoid. c. 0.22 A d. 2.2A...
5 answers
On the blanks next to each beam, predict what type of radiation is represented:Lead blockBmyT4a0IERadioactive EunshnceElectrically charged platesPhotographic plate
On the blanks next to each beam, predict what type of radiation is represented: Lead block Bmy T4a 0IE Radioactive Eunshnce Electrically charged plates Photographic plate...
5 answers
Find h'(t) if h() = 8.2 - 5.5t + 0.8t2 h () =
Find h'(t) if h() = 8.2 - 5.5t + 0.8t2 h () =...

-- 0.022374--