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For the matrix A below; find a value of k so that A has two basic eigenvectors associated with the eigenvalue ) = -2-1 1 -4 11 -2 -2 A = 0 -3 k 0 -2k= 0...

Question

For the matrix A below; find a value of k so that A has two basic eigenvectors associated with the eigenvalue ) = -2-1 1 -4 11 -2 -2 A = 0 -3 k 0 -2k= 0

For the matrix A below; find a value of k so that A has two basic eigenvectors associated with the eigenvalue ) = -2 -1 1 -4 11 -2 -2 A = 0 -3 k 0 -2 k= 0



Answers

find the values of $k$ for which the matrix $A$ is invertible. $$A=\left[\begin{array}{lll} 1 & 2 & 0 \\ k & 1 & k \\ 0 & 2 & 1 \end{array}\right]$$

All right. We find, we want to find the values of K. For which the matrix A. Is convertible. Where A has rose K two and two. K. A is simply a two by two matrix given here in the upper right of the document. So in order to answer this question, we need to be able to take the determinant of K. Or rather disturbing enough Matrix A. And identify from that determinant where A is not convertible to do so we make a note of the fact that the determinant of a matrix A. B C. D. Is A. D minus C. B. And that matrix matrix A is inevitable if and only if the determinant does not equal zero. So we need to solve for the determinant and identify for what values of cable determinant does not equal zero. So to start off, the determinant of A is simply keychains K minus four or k squared minus four. We solve that for zero. Sochi experiments for equal zero, which reduces the case critical for or K equals plus or minus two. When the determinant zero, so monkey equals minus two, the determinant zero Ak is not inevitable, so that means that A is convertible whenever K does not equal plus or minus two.

We want to find the value of K. For which the matrix A. Is inevitable with A is given by Rose k minus three, negative two and negative two K minus two. So remember the determinant? About? Two by two. Matrix with a. B. C. D is simply a. D minus C. B. And that matrix A is in veritable if and only if the determinant is not equal to zero. So we want to solve for the determinant of A and then determine for what values of K. This determinant does not equal zero. Such a Matrix A is convertible. So first determine the way is just k minus three times minus three minus 16. We set this equal to zero so k minus three times minus 16 equals zero or k minus three times came last week was 16 which gives K is equal to 18. K is equal to 19, so K is equal to 18 and 19. When this determinant is equal to zero. We want to find the values of K for which A. Is not convertible. So we have that. All values of K excluding 18 and 19 allow major K. Two convertible or A is in vertebral, as long as kate is not equal 18 or 19.

Idea for this question were given Matrix eight on the edge of values and workers off it, although do not know the Matrix itself on then we know the area solution off a difference equation on Dhere For the first question, we're going to compute acts one because eight times at zero, I wouldn't know. Oh, here, let's just set a to be a B, c and D since it is two times two matrix Astro s So we have this Diaco's too well, no, I a plus B and I see hoops fast be mocha. This is a So this got here is our X one and we're going to compute it. So that's what we want to the information. We already have things. We know that for metrics a It so satisfied this equation it shouldn't be. It should be. I'll be here for our angel lepers eight. House, we because Lambda comes be So this is our that initial for it on values and entering batters on Let's just right up. So here it should be a B C and the palms V for first be it is 11 and it equals two Lambda Tom's V O, which is three and three. It is very obvious, Onda, for our next equation, we will have A, B, c and D. And here we're gonna use the second Adam Factors, which I think should be minus 11 here on the equals to one of those three Toms O V two, which is Nega to 1/3 and 31. Overstreet. Okay, so what do we get for this? Let's just kind of rewrite it. We know that here for this part, because to a plus B, uh, then C plus D. Okay, uh, you Khowst 33 And here, similarly for this part is not It's in a B for this part is now that you've seen a plus d equals negative one or three and then one or worse street. OK, so actually we can compute a, B, c and D according to these four equipments, because because here we have foran noise and Warrick Regis wolf. But actually we do not really need to compute a, B, c and D. And here that's because we only need to know night a plus B at Nice a plus B a trustee in older the computer that's one. So that's just, you know, this could interview one than this to be true. And let's see how, for example, I want to use a plus B minus A plus B and combined is two to get not eight months. So how can we do that? It's actually quite easy. Let's just use five times one, uh, minus war comes to let's see what we got. Uh, nine times a minus four grams minus times minus a gives us night. And here for B, we have five B minus four b ecause speed. So we get exactly not a plus B. So we have, uh, nine a plus b because, uh, street cams file, which is 15. And here I run it as 45/3 them minus four times 91/3, which is plus four, or street. So it gives us, uh, 49 or street, okay. And for our second 2nd 1 element off these weapons, we have nicely plus city on five times one minus four times the second ingredient. Also give us a nice C plus d equals to file come street, which is 15 which are right here as what if I over three the minus four times one was three. So it here I write here as four. Illustrate minus roast. So it gives us street. Uh, okay. 41 Overstreet 41. The worst. That here is 14 hours. So way have solved the first problem, which is to get x one. So acts one Here is 40 night or three and 41 or three. Okay, so that's We'll launch our second problem, which is here. We want to compute axe. Okay, let's see how to do that. Well, we already know XK because to eight humps x k minus one. But it's not enough here. We only know x zero. Right? So we need to write it or at or it's like a recurring process. Soul. Let's do it. Well, right. XK minus one as eight times a x Q minus two. Or, uh, or that's just write a to the power off to hear. All right. Okay, so we have a to the power of three hubs, x k ministry and so on and so on and so on. Do it. Finally gets a to the power off K ax there. Since at zero, we already know, right? At zero. It is ny one we already know. Accidental. So here we only need to compute eight to the power of K in order to get access came. Okay, let's see how to do that. As we know, they has to a distinct and rail vaginal values three on one or three. Right? So we know that a is die ago, Nall Izabal. So in other words, we can rewrite ax to the pharaoh minus one times a to the Times X because the and here acts is, uh well, that right here is This is the first Adam Wachter is the second panel doctor. And so off a on for D, it is ah, Matrix comprised off all aging values away on it is the diagonal matrix. So here it is. Lump one long up to and so on. A racing else zero. Okay, so here for this we can also rewrite it as a equals two AKs, the AKs, English. There we get a to the power of K because to, uh, axe Tom's d to the power of Kate Times x inverse. It is Because, for example, if we want to compute a to the power of to then we simply have axe English times acts. Then these two term cancels out and we have a the tooth east combined together. So we have X times d to the power of two times actually were. So it is the same for all of the other order. No meals off a. So we have eight of our Okay, it goes to this quantity. And that's now compute ate the problem, k. So it is Attari equal to ax here. We already know it is 11 not the one on one times d are okay, Well, it is every, uh quite either. Here it is ST to the power of K one or street for Okay. Since it is a diagonal matrix, we have everything else that goes to zero. So here it is. Two terms of mixtures. In this case, we have another one 111 the worse. Okay, so the only thing remains to be done is to compute this quality, which is the anglers off this matrix? Well, since it is a two times two matrons, it is not that complicated to compute the inverse off. Mr. Well, let's just use. Uh, this way. The compute, for example. That set this quantity to be, for example, a one C one. Do you want the war? And And when it comes 11 Now, if you want one, it must decode to identity matrix, which is 1100 So here is this quantity is exactly the thing worse off. Uh, this matrix here, Right? And after, uh, computing the corresponding terms for anyone. Dont want anyone. Let me just give you the results off these quantity or this matrix equals to 1/2, uh, minus one. No. 1 to 1 or two and one or two here. So we have, uh, XK Okay. Yuko's, uh, age the part age, the ROK comes back zero chase. Let me just roger Stone 1111 three to the power of K one or three to the four K than their own hero pumps. Uh, one or two narrative one or two lower to one or two. Then Tomczak 00 is 91 Okay, so here a racing is known, we can use this one here to compute a couple. The results we need and it turns out, uh, that's XK equals to fly. Homs three you could part of K. Then comes 11 minus four palms one or three to the power of K. In times negative 11 Here we know that 11 is our We want no 11 is over. Be too. So we have expressed at the part of K with okay, We want on the way to here. And it can be obtained from by computing this chronic because we have a racing we need for for this column Here it is. We want Just call him here is if we two we have k here and everything else is no already. So we can't get this quantity. And that's all for computing at the power. Okay. Thank you.

This problem asks ist friend Eigen. Values of the Matrix A with the given Eigen vectors could do this using the equation. A times V is equal to Lambda Times e So first for the first Eigen vector V one, you can plug in a 4213 then V one vertically. One negative too is equal to lambda Warn times one negative, too. Then doing matrix multiplication on the left side, we got the top row to be four times one, which is four plus one times negative two, which is negative two and two times one which is two and three times negative two, which is negative. Six is equal to to negative four and that is equal toe lambda one times one negative too. So we could see the constant Lambda 11 Dividing to divide by one and negative four by the bynegative two gives us land. One is equal to two. Then for the second Eigen vector view, too, we can plug in again a 4213 Then the vector V two wishes 11 is equivalent to lambda two times 11 again matrix multiplication of the left side, The top row. We get four times what it's for. Last one times one, which is one 1212 plus three times one is three, which gives us 55 which is equivalent to land a to times 11 that if we divide five Vector 55 by the Vector 11 We find that landed too is equal to five, and those are two Eigen values.


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