5

For the matrix A below, find a value of k so that has two basic eigenvectors associated with the eigenvalue / = 2.-2 472 ~20 0 -k=...

Question

For the matrix A below, find a value of k so that has two basic eigenvectors associated with the eigenvalue / = 2.-2 472 ~20 0 -k=

For the matrix A below, find a value of k so that has two basic eigenvectors associated with the eigenvalue / = 2. -2 4 72 ~20 0 - k=



Answers

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

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.

We want to find the values. Okay, for which matrix A is convertible, matrix A is a three by three. Matrix throws 1 to 0 K. One K. M 0 to 1. As shown in the upper right. To solve this problem, we're gonna use the relationship between the determinant and matrix convertibility. Remember that to find the determined about three by three matrix with rows A B c, D e, f g h I. We use the formula here on the left. Then we can use the fact that A is inevitable if and only if it's determined does not equal zero. So we want to find where from the determinant formula. Certain values of K. Make are determined zero and go from there. So first applying the formula, the left, the determinant of A is one times one minus two K minus two. James, t minus 00 or two minus four K. So solving for zero, we have two minus four K equals zero or 14 equals two or k equals one half. That means that are determinant is zero and that is not inevitable when K equals one half. So we must have that Matrix A is convertible whenever K does not equal one half.

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 gives us a matrix and asks us to sell for its guiding values and Eigen vectors. We do this first by finding the characters to polynomial, which is found by taking the determinant of the matrix a minus lander times the identity matrix. This is going to give us the determinant of the matrix with negative land all along of diagonal and then to everywhere else. Something for this will give us opponent. You don't Negative land. A cute plus 16 plus 12 Landau, which is equivalent to negative Lambda months to Lambda minus four. Lead two plus two. We're gonna set that equal to zero. That gives us our Eigen values to equal for and negative to where Negative two has an algebraic multiplicity of to So in order, solve for the Eigen vectors, we have to take a minus land of times. I times I director ex national equal the zero factor. So first, with the four eyed in space, we plug this in, and that will give us the Matrix Negative. 4 to 2 to negative for two to to negative four times x one. And then I'm gonna divide um, every road by two expected performed Yashin elimination and just make this solving easiest. We get negative. 211 one night of 2111 negative too. Times X one equals the director. That means that negative to a one plus B one plus C one equals zero. I want minus to B one plus C 10 and a one plus B one minus two. C one. It was zero when we sold for the system of equations. We will find our wagon vector X one, 111 Then we do the same for Lambda equal to negative two. Doing this will give us the Matrix, uh, with twos in every row. So there we go times the director X two, and then we can perform gashing elimination. Um, by subtracting the second subtracting the first row from the second and third rose and also divided by to give us 111 and then zeros everywhere else. And that should equal the zero factor. In doing so, we can get to major cities that span this, um, argon space and those air going to equal negative 101 and night of 110 as they're too linearly. Independent solutions. And that's our final answer


Similar Solved Questions

5 answers
4) (10 points) Let flx) = {ax + b if x >-1 bx2 _ 3 if x <-1 Find a and b that make flx) differentiable everywhere. 661) -3 = b~ > 7 9+6 06-0+5 43>
4) (10 points) Let flx) = {ax + b if x >-1 bx2 _ 3 if x <-1 Find a and b that make flx) differentiable everywhere. 661) -3 = b~ > 7 9+6 06-0+5 43>...
5 answers
8 a) SnOz crystallizes in a rutile structure (see Table 7.4). Using the equation we developed in class, determine the theoretical value for the lattice energy of SnOz: Ionic Radii: Snt 83 pm; 02 = 126 pm. Assume that the Born Exponent for Sn is 10.b) Compare your answer for the lattice energy of SnOz to the answer you got above in problem Sa for the lattice energy of KBr: Which value is higher? Why?
8 a) SnOz crystallizes in a rutile structure (see Table 7.4). Using the equation we developed in class, determine the theoretical value for the lattice energy of SnOz: Ionic Radii: Snt 83 pm; 02 = 126 pm. Assume that the Born Exponent for Sn is 10. b) Compare your answer for the lattice energy of S...
5 answers
[2 marks] Using the Taylor Remainder Theorem what is the upper bound on If(x) T4(x)l, for x €[4,10] if f(x) 2 sin x and T4(x) is the Taylor polynomial centered on 7_
[2 marks] Using the Taylor Remainder Theorem what is the upper bound on If(x) T4(x)l, for x €[4,10] if f(x) 2 sin x and T4(x) is the Taylor polynomial centered on 7_...
5 answers
Of the substance? Show all How many moles and molecules of COz are in 100.0 using the correct number of significant figures- work Report your fina ansvers (10 points)
of the substance? Show all How many moles and molecules of COz are in 100.0 using the correct number of significant figures- work Report your fina ansvers (10 points)...
5 answers
What value of Zol2 in the CI formula results in a confidence level of 75%?
What value of Zol2 in the CI formula results in a confidence level of 75%?...
4 answers
(10 points) Let be the relation on Z+ given by € y whenever wly"_ For each "reflexive, "symmetric; "transitive;' determine if satisfies that property. If "yes YOl need not prove (just say "ves' If "no. justify why with an explicit example.
(10 points) Let be the relation on Z+ given by € y whenever wly"_ For each "reflexive, "symmetric; "transitive;' determine if satisfies that property. If "yes YOl need not prove (just say "ves' If "no. justify why with an explicit example....
5 answers
R=0"Given that lim % =! = 1, find f"(0) for f(x)=otherwise:Rolle' s Theorem: If f is continuous on |a, b] and differentiable On (a, b) and if f(a) f(b), then there is point in (a, b) for which [' (c)The Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on f(b) f(a) (a,b), then there is a point in (a, b) such that f %(c) =
r=0 "Given that lim % =! = 1, find f"(0) for f(x)= otherwise: Rolle' s Theorem: If f is continuous on |a, b] and differentiable On (a, b) and if f(a) f(b), then there is point in (a, b) for which [' (c) The Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiabl...
5 answers
To understand the strength of a relationship between two variables calculate aslopeCorrelation coefficientpredicted value of ymodel
To understand the strength of a relationship between two variables calculate a slope Correlation coefficient predicted value of y model...
5 answers
By about how much will g(X,Y,Z) = 2x+Xcosz - y einz + change if the point P(x,Y,Z) moves from Po(1, _ 2,0) distance of ds = 0.2 unit toward the point Pa( - 1,0,2)?(Type an integer or decimal rounded to four decimal places as needed )
By about how much will g(X,Y,Z) = 2x+Xcosz - y einz + change if the point P(x,Y,Z) moves from Po(1, _ 2,0) distance of ds = 0.2 unit toward the point Pa( - 1,0,2)? (Type an integer or decimal rounded to four decimal places as needed )...
5 answers
Determine whether or not each structure is a carbohydrate. If the molecule is a carbohydrate, classify it as a monosaccharide, disaccharide, or trisaccharide.
Determine whether or not each structure is a carbohydrate. If the molecule is a carbohydrate, classify it as a monosaccharide, disaccharide, or trisaccharide....
1 answers
Find det $(t)$. (a) $T: R^{2} \rightarrow R^{2},$ where $T\left(x_{1}, x_{2}\right)=\left(3 x_{1}-4 x_{2},-x_{1}+7 x_{2}\right)$ (b) $T: R^{3} \rightarrow R^{3},$ where $T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}-x_{2}, x_{2}-x_{3}, x_{3}-x_{1}\right)$ (c) $T: P_{2} \rightarrow P_{2},$ where $T(p(x))=p(x-1)$
Find det $(t)$. (a) $T: R^{2} \rightarrow R^{2},$ where $T\left(x_{1}, x_{2}\right)=\left(3 x_{1}-4 x_{2},-x_{1}+7 x_{2}\right)$ (b) $T: R^{3} \rightarrow R^{3},$ where $T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}-x_{2}, x_{2}-x_{3}, x_{3}-x_{1}\right)$ (c) $T: P_{2} \rightarrow P_{2},$ where $T(p...
1 answers
Find each logarithm. Give approximations to four decimal places. $\log 457.2$
Find each logarithm. Give approximations to four decimal places. $\log 457.2$...
5 answers
If the drum with radius of 100mm and it rotatesthrough 8.5 revolution. Find the following: a) total angle through which the drum rotates in degreeb) the total angle in radianc) the length of the rope that has been wound on to the drum
If the drum with radius of 100mm and it rotates through 8.5 revolution. Find the following: a) total angle through which the drum rotates in degree b) the total angle in radian c) the length of the rope that has been wound on to the drum...
5 answers
What is the product of the following reaction sequence?NHz1. Hyot 2. HzINiOHNHNH
What is the product of the following reaction sequence? NHz 1. Hyot 2. HzINi OH NH NH...
5 answers
9 D=l0E-[0 %] +.J a:J 1Is the matrix F in Problem #1 above invertible? Why or why not? If so, find F-
9 D=l0 E-[0 %] +.J a:J 1 Is the matrix F in Problem #1 above invertible? Why or why not? If so, find F-...
5 answers
Gbulgie soluina 04 trar nendnd Oregaa 10.250 64 conion witn 22 40 KNOJ10i,1 grrncn:
Gbulgie soluina 04 trar nendnd Oregaa 10.250 64 conion witn 22 40 KNOJ 10i,1 grrncn:...
5 answers
Consider this net or overall reaction XY +Z X+YZWhat is the rate law for the reaction based on the proposed mechanism below? XY+ XY XY2 + X Ik1; slow XYz + Z XY + YZ kzi fastDlrate k[XIIZ]Orate k[XY 2JIX]Orate kxYJIZ] Drate # k[XY]?
Consider this net or overall reaction XY +Z X+YZ What is the rate law for the reaction based on the proposed mechanism below? XY+ XY XY2 + X Ik1; slow XYz + Z XY + YZ kzi fast Dlrate k[XIIZ] Orate k[XY 2JIX] Orate kxYJIZ] Drate # k[XY]?...

-- 0.025648--