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Considcr thc matrix:Compute all eigcnvalucsandl thcir eigenvectors ViMATk?Give Ilalrix F thal diagoualises Vcrify vour choice:Lo DLhal p-'APmarksl2019 Semester...

Question

Considcr thc matrix:Compute all eigcnvalucsandl thcir eigenvectors ViMATk?Give Ilalrix F thal diagoualises Vcrify vour choice:Lo DLhal p-'APmarksl2019 SemesterAssignmnentPage of ??(iii) Find te veelor A'5'7 IIINT: Writelinear CUnbinalion U FUUEcigcnvertors, thcn Use thc lincarity propcrty on A"(avt Ivz} . matk: A verlaill COLIQily in Aucklanl serviced by 3 superarkel slres A B & C. Assume the community population stable and that Cinstomer and do change to ditferent superm

Considcr thc matrix: Compute all eigcnvalucs andl thcir eigenvectors Vi MATk? Give Ilalrix F thal diagoualises Vcrify vour choice: Lo D Lhal p-'AP marksl 2019 Semester Assignmnent Page of ?? (iii) Find te veelor A' 5'7 IIINT: Write linear CUnbinalion U FUUE cigcnvertors, thcn Use thc lincarity propcrty on A"(avt Ivz} . matk: A verlaill COLIQily in Aucklanl serviced by 3 superarkel slres A B & C. Assume the community population stable and that Cinstomer and do change to ditferent supermarkets; February Ist study foutl that 1/4of thc Comuity shopped at store 1/3at store B; #nd 5/12 at store C. It Was also found that each month On aYerage store rcrained] 90% nf its customcr basc. but lost 10% to storc R_ Srore B reraincyl 5% of its cuslomuers bul lusl 85% u llem Wu slure ald 10% ol them slore C Slure relainexl 407 of its customcrs losing 50 tn storc and 10% to store B_ Wiite the transition matix P for this Markov PIOCCSS marks] Whal pruporlion O cuslOlers will each slure relain by Mareh Isl? [arks] I lle KAM pallern lomer dislribuliun cOnlinues wlal is (lle expecled Jong lert dislribuliuu o cusloqers amuwg lhe llree slores? [ark:l



Answers

(a) Show that the matrix
$\mathbf{A}=\left[ \begin{array}{ll}{1} & {-1} \\ {4} & {-3}\end{array}\right]$
has the repeated eigenvalue $r=-1$ and that all the eigenvectors are of the form $\mathbf{u}=s \operatorname{col}(1,2)$
(b) Use the result of part (a) to obtain a nontrivial solu-
$\quad$ tion $\mathbf{x}_{1}(t)$ to the system $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$
(c) To obtain a second linearly independent solution to $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$ try $\mathbf{x}_{2}(t)=t e^{-t} \mathbf{u}_{1}+e^{-t} \mathbf{u}_{2}$ [Hint: Substitute $\mathbf{x}_{2}$ into the system $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$ and derive the relations
$$ (\mathbf{A}+\mathbf{I}) \mathbf{u}_{1}=\mathbf{0}, \quad(\mathbf{A}+\mathbf{I}) \mathbf{u}_{2}=\mathbf{u}_{1} $$
since $\mathbf{u}_{1}$ must be an eigenvector set $\mathbf{u}_{1}=$ $\operatorname{col}(1,2)$ and solve for $\mathbf{u}_{2} . ]$
(d) What is $(\mathbf{A}+\mathbf{I})^{2} \mathbf{u}_{2} ?$ (In Section $9.8, \mathbf{u}_{2}$ will be identified as a generalized eigenvector.)

This problem gives us this matrix and acts us to solve first Eigen values and Eigen vectors. You do this first by finding its characteristic polynomial, which is found by taking a determinant off the Matrix minus lambda times, the identity matrix that will give us the determinant of the matrix five minus slammed, uh, along the diagonal and zeros everywhere else, which I'm just gonna leave blank those roll zeros that gives us the polynomial five minus lambda cute and setting. That equal to zero gives us that our Eigen values equal to five with an algebraic multiplicity of three due to the Cube. So then we'll use land equals five. So plug this into a mind slander times I to find the I director X That gives us zero factor. So playing in land equal to five, that will give us any tricks with zeros everywhere. So, as a results, we will have in order sulfur this system. Um, this wagon space will span three vectors, and those three vectors will be our Eigen vectors. They're 100 010 and 001 of which are literally independent from each other, giving us that since our final answer

This problem gives a matrix and asked us to find Eigen values and Eigen vectors. We do this by finding characteristic polynomial which is equal to the determinant of a minus. Lambda I and set it equal to zero. That will give us the matrix one minus lambda 00 00 minus land a negative 101 negative. Lambda, Take the chairman of that will give us, uh, lambda squared times one minus lambda plus one minus lambda, which simplifies to Lambda squared plus one and one minus lambda. And we set that equal to zero, which, give us gives us that our guys are equal to one and plus or minus I In order to sell for the Eigen vectors, we need to find vectors X such that a minus lamed I times that Vector X is equal to zero vector. So first, starting with land equal to one we plugged that into a minor, slammed I and it will give us the matrix. 000 zero Negative one negative. One 01 negative. One times the erector x one. We can do gash elimination or to simplify the calculation. So what I'm going to do is add the, um, second line to the third line that gives us we'll have all zeros on top. Still zero night of 11 and then we have zero and negative 20 times x one which dividing by two on the bottom row and dividing Yeah, divided by two On bottom row, we saw zero here zero night of 11 zero 10 times X one And then one more step. I'm going to add the, um, bottom road to the second row. Sweet balls, years on top. Then 001 and 010 times a one B one C one should equal 000 Since here we see that B one and C one have to equal zero. Then we can find that are Eigen. Vector X one is equal to 100 Next, we'll do land equal to five. But land equal toe I we get the matrix one minus I 00 zero Negative. I minus one and 01 modest. I times the Eid wrecker Vector x two. We can simplify this. I'm going Teoh, multiply the, uh, second row by I and add it to the third row. That gives us one minus I 000 minus one. And on the bottom row, we have all zeros. Times A to B to C two should equal 000 then solving this system. See that a two has two equal zero and that ah b two is going to have to equal Ah, odd or so items B two is gonna equal C two. So that will give us our Eigen vector x two to equal zero I want Lastly, we're gonna do land equal to negative. I playing that in to a Muslim, I we get one plus I 00 zero I negative 101 I times x three again performing gash In elimination We can multiply the second line by I and subtracted from the third row, and that gives us the same on top seeming second round. And then we have all zeros again on the third row times a three b three C three, which is equal to 000 and solving for this system, we see again that a three has to equal zero and then we can find that B three is equal to negative. I and C three is to one. And that will give us our third Eigen vector, which is equal to zero Negative. I won, and those are our final answers.


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