use-the-factorization-ap-d-p-1-to-compute-ak-where-k-represents-an-arbitrary-positive-integer-left-2

Question

Use the factorization $A=P D P^{-1}$ to compute $A^{k},$ where $k$ represents an arbitrary positive integer.
$\left[\begin{array}{rr}{-2} & {12} \ {-1} & {5}\end{array}ight]=\left[\begin{array}{ll}{3} & {4} \ {1} & {1}\end{array}ight]\left[\begin{array}{ll}{2} & {0} \ {0} & {1}\end{array}ight]\left[\begin{array}{rr}{-1} & {4} \ {1} & {-3}\end{array}ight]$

Bobby Barnes · Accepted Answer

So if we want to raise this to the K through power, let&#x27;s just go ahead and do this really quickly. So first, remember, this is supposed to be a is a good P D. P in verse. So if we were to kind of raise this to the case of power on each side So Okay, okay. Remember, we can&#x27;t just distribute the K Tau everything. We&#x27;d have to do this P D p Inverse times PDP inverse times PDP inverse Over and over again. So that tells us the PM PM versus will all cancel except for the ones on the end. So then we would have over here negative to 12. Negative. 15 raised to the Kate is going to be so we just have p first. So 3411 And then we have this middle one where it is raised to the k. So 201 race, the k and then we have our outside once a negative one for one negative three. And then remember, multiplying this one mental is a lot easier since all we do is take that K and then apply it to the diagonals. So be 3411 Then we get to to the K 001 to the K, which is just one. And now it just turns into multiply these matrices together. So let&#x27;s do the first one over here. So we first would dio uh, three times two case that would be 32 k and then four times zero. So that&#x27;s just going to be that. Then we would do 34 times, 01 So that just leaves us with four here. Then we would do 11 times two k zero. So that&#x27;s just two K. And then we get zero there. Right? So that&#x27;s the first, uh, zero. Because we still need to this last one. So 11 time 01 which gives us one, and then we still have our matrix over here. So negative one for one negative three. So now we do again. The first row, First column. So 32 k times night once. That would be negative. Three times to K and then four times one which is going to be plus four. Then we do same one, but for negative three. So that will be so. Four times three is going to be 12. So this would be 12 times two K and then four times three negative 12 and then we do two k times. Negative 11 So that would give us negative two K plus one and then forward last one, we&#x27;d get so two K times four so four times to UK and then one times three is negative three. And so this is what we&#x27;re going to get if we were to x raise that original matrix to the case Power Assuming again K is just some positive integer.

Samuel Goyette · Answer

Okay, so we have once again, uh, in a dining dia generalization of the matrix PDP minus one. What we want to know is a general formula for a to the power case of what is a in the power k for Okay, uh, positive integer. So we know that a que is p times d to the power k times p minus Juan based on page 284 of the boat. So we&#x27;ll just now we plug in the values of P and G, which are given a question. So he is 1031 de is a 00 B. So a diagonal matrix, two of their power K and P minus one is 10 minus 31 So PND are given now because he is a diagonal matrix. What can we do? We can simply ah say that d to the park is a k agent of power K 00 b to the power K because he is a diagonal matrix, P and P minus one don&#x27;t change. So we just justify what we do because the is ah, diagonal matrix. Perfect. Now what do we do? We just compute this matrix product. So we start with the two matrices on the left. So here we will have a do that power K zero three A to the power K B to the power K. So we just doing normal standard two by two matrix multiplication. And we have P minus one. So 10 minus 31 and we do one more matrix multiplication. So we&#x27;ll have a que zero three A k minus three b to the power. Okay. And I will just be raised zero and make sure everything is perfectly allying and we&#x27;ll have B k right here. So the and that&#x27;s the final answer. So for Okay, positive integer We have a K agent of power K. It is a small age of the power K 03 small age of the power. K minus three. Small beating the power K. Yeah, not zero b. K. And there is your final answer. So what it we used the main theory in to use in that question is the fact that a did it power. Okay, is p times D to the power K times p minus one for any diagon ization of the Matrix

Ankit Gupta · Answer

Okay, Head. In this case, the value of K is given toe on B X is defined US two X cubed minus three X squared, minus five X and bless for Okay, so we just need to find the value off. Be off, right? Do not be off. Cable was simply by using this Internet division processes to right here on the coefficients are two minus three, minus five and four. OK, the stories here as it is. So this is to okay No, 212 times this for right to simply consult before on this is what the difference is 1 to 11 times is to the differences minus three and 21 minus three times minus six Differences. My sister with this is that particular value. Unless we can say that the value of P off to R P off days equals two minus two. All right,

Ankit Gupta · Answer

we have been given that the p X is equal sto X cube minus two X is quiet minus seven X minus four and gay his e questo minus one. Okay, call to minus one is one of the zeros off this P X. It means clearly that X plus one is one off its factors. X plus one is a one off the his one off the factors. Okay, so if it&#x27;s one of the factors, we can divide it. But your flex can be divided with that. But the thing that is X plus one and you can leave you remained a zero Sylvia, your value as galactic. There you go first with X squared, I guess you excuse. And they could give you plus x esquire. So, after moving that will be left with this particular cancel soldier and discover minus three X squared minus seven X minus four and no, after a thorough, whom they will get minus X. So if we get minus three, it will be left with minus three Xs square. And here we get minus three x no fear the practicability minus four x minus school. Hang in There. You get minus one minus 4 40 because you need to get over that minus four. Exactly. So we&#x27;ll go with minus four for go with minus for illegal miners for X minus four, and after selling it, it&#x27;ll become zero. No, we have backed over two weeks. Is nothing work? Excess squad minus three X minus. Food forms for crazy looking, vague excess squad, and we can go with minus four X Plus X minus four. Now we can take X common will get explainers for that plus fund. Common, we&#x27;ll get X minus four. Finally, your factor becomes X plus one, multiplied by X minus. How these two factors are nothing but your practice off your ex and if you want represented PX in the form off linear factors, it means you need to get the product off. It&#x27;s all zeros, so one of the zeros is clearly given in the question. As your X plus one and other two are being formed like this to express one multiplied by again X plus one on multiplied by X minus four. And these are the Lena Factors off the beings

Ankit Gupta · Answer

Okay, Head in this case that I left is given square with three. And we x is defined us export for les two x cubed. Okay, this is two X squared. Right, So this is two x squared on minus 10. All right. And we just need to find that I love be off using synthetic division process of this is room three on here in the skins is 12 and minus three is equals to 1.7. If I just approximately this value crazy equals to 1.7. So this value simply comes out, people disparately So when in here. So this time this is 1.7, So this simply comes out with 3.7. Okay, now, this values approximately 1.7 to 1.7 times 3.7. All right, So that the 6 29 6.29 And if I just make a defense in here, This is fun. Okay, Now eight minus 27 And this is nine minus six, which is so this is the particular value. And hence we can say that therefore, be off square. Wrote off three is equal to three point seven, but all right,

Ankit Gupta · Answer

we have been given that P x go to X Q minus two. Texas car minus five. Explosives and Casey questo. One is one off zeros Know if case goes toe. One is one of her sealed. It means X minus one. Must be It&#x27;s his up in fact, it off speaking. So if x minus one is affected, let&#x27;s divide this particular pick You then a p x Sorry. We get two X Q minus three X squared minus LILEKS plus six. And we will do adding it with your X minus one. So what, we get regretted to access Square becomes toe Excuse and Eric it minus two axis. So it cancels out on display spirit minus X squared minus five x plus X and know what begin toe. They can go for minus X, so it gives you minus six s squad and get get super sex here. Also, you will get minus, uh, six x and blessings. If you subtracted and it lasted what we can do, you can go with minus six. And if you go with minus six, it will give it a value of minus six X plus 600 vanishes. What we have got. We have got that X caution is nothing but looks a square minus X minus six. And we need to uh, yes, Fertilize it. So we confer. Cries it with the help off. Minus four X plus three x six. Now we can take two x common. We get X minus toe, we can take three. Government will get X minus two. Are this case US toe explicit three on another record gives X minus toe frequent. This particular thing is really ineffective off ux. No. If you need to re present what be X we can easily represent or PX in the terms off factors lena factors, that is X minus one. According particular question on the coup that were just called with two exits to explosively and X minus toe. Hence, if you represent BX in the form of Pristina, factors little with something like X minus. One will depend, but to explicitly multiplied by X minus two

Bobby Barnes · Answer

Okay, So have a K. Is this e quickly to V. P? Okay, Pete Inverse. That&#x27;s equal to or even a few. Which is 3411 to UK zero. So one. Okay. And he invested through that one for one in 83 that simplifies still or ministry times. Okay, 12 times to UK My swell. They could have two K That&#x27;s one and four times too. Okay, my mystery.

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The matrix $A$ is factored in the form $P D P^{-1…

Problem 4 Easy Difficulty

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