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Find the matrix to which P" converges as n increasesThe matrix converges to (Type an integer or decimal for each matrix element: Round t0 fve decimal places as...

Question

Find the matrix to which P" converges as n increasesThe matrix converges to (Type an integer or decimal for each matrix element: Round t0 fve decimal places as needed )

Find the matrix to which P" converges as n increases The matrix converges to (Type an integer or decimal for each matrix element: Round t0 fve decimal places as needed )



Answers

Find all values of $p$ such that the sequence $a_{n}=\frac{1}{n^{p}}$ converges.

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Hello there. Okay, so this occasion we've got this matrix A. That you should not just these uh up triangular matrix. So that is going to help us a lot in the process and we need to die agonizing metrics. That means we can write this matrix A as a table matrix. The that is going to be equal to the inverse of some matrix B times A times B. Okay, so that's the process of the organization. And the way to do it is to first calculate the Eigen values and then compute the Eigen vectors. So the matrix D. It's going to be developed. Matrix were in the diary, you will find all the values and in the again, factors will compose these metrics speak, that is going to obtain it's going to be composed as the Hagen pictures as combs of these matrix. Yeah. Okay, so that's the process of their association and that's why what we're going to do right now. So let's compute the Eigen values and that is actually really easy in this case because as I mentioned before this an upper triangular matrix, that means that the Eigen values are on the diagonal. So the alien values for these matrix A. It's going to be too And they can value two and 3 are going to be equal to three. Actually, it's just one I can value with multiplicity. So let's calculate the vectors are associated to these values. Okay, so I'm the one equals to two or in this case is to what's the Eigen value for this? So that means taking a minus lambda, the identity matrix and fight the solution for the system. So that is equal to 00 highness too. 030 here is one And 001. Yeah. Great. So we got the system X, y Z equals to the zero factor. So let's find a solution for the system. You can notice that this system has a free variable because we can eliminate one of this. Uh one of these uh Can be written as a linear combination of the other two. What I mean is for example, here the first role is a linear combination of the 3rd 1. So what we obtained here is 000 010001 X Y. Z. I don't need to find this asian for this system. That is yeah. To be one, it's going to be just so uh X can take any value because well the solution for the system is that X can be any value and Y and Z should be equal to the, So the Eigen vector updated to this Eigen value here is one zero zero. Great. So We got that the first Eigen value is equal to two and the associated Eigen vector is equal to 100. Now, let's continue. We got that. The 2nd and 3rd Eigen values equals 23 of which is the same. They have is uh and they can value with multiplicity too. So we should find to Eigen vectors for this system. Okay, so the matrix a mine Islam. The two I X equals to zero. We should find a solution for this system. So in this case I'm going to copy here the matrix again. So it's 20 minus 22 or 30003 Okay, so this matrix after subtracting the matrix times the Eigen value, We obtained 10 -2000000. So the solution for this system Is X -Z equals to zero. And why is a free variable in this case? So let's say that Y is equal to a for example. And from this we got that X should be equal two times E should we constitute times? See so the solution is equal. Uh huh. Two to to hear. So here we got another free variable. That is easy. So if the is it goes to be, we have the following solutions. So two times B. A. And here be yeah, this is the solution. So we can separate this solution in to that is equals to eight times 010 plus B. 201 And these two here The 01, zero and 201 corresponds to the Eigen vectors two and three. Right? So at the end we got that for the first I can value equals to two. We got the Eigen vector 100 And for the again value to three equals 23 We got the iron vector 010 And the Eigen vector 201 Great. So these are the Eigen vectors. And let's construct these demetrius is so the matrix is going to be 200030003 And the matrix B. It's going to be 100 and uh 010201 So maybe you're wondering in what order we're going to put these. I can vectors here. Well, the order doesn't matter in this case because these two Eigen vectors are have multiple attitude. So we need to put the Eigen vectors associated to this Eigen value in the position. So this case these repeated times. So we need to put one of the again vectors in one column and the other in the other one. This doesn't matter the other of these two. Great. So now we need to check. So that means taking the embers of these Matrix times A times B. So the members of this matrix P. Is well we need to check that this is because today but one other way to calculate this is taking multiplying by P. To do to the left. And that means that It is enough to show that 18 speed is equal to p times deep. Okay so P times D. Is equal to 102010001 Times 200 030 and 00 three. This is equal I want to check this. Yeah. Okay So this is equal to two three three two three three and zero 03 If I'm not from. Oh sorry yes you're good mistake. So it is too three. Um Three. Yes. And the 2nd row is zero three zero. Yeah that's it. Uh huh. And you can actually calculate that a the Easy constitute 0 -3. You're 30 and 03 times The P Matrix 020 1000 1 is equal to 233030 and 003 So these two matrices are the same, so this is that the organization of this matrix A.

There. So for this occasion we got this matrix A. And we need to dig analyzes matrix. That means we need to find this to me to see here, D and P. So the process of the organization means that we can find some metrics P. Which is veritable such that after we multiply in this way with a we obtain a diagonal matrix. So the process of the organization consists on first calculating the Eigen values and these are going to come both the diagonal major. So in the day, well we're going to put the Eigen follows and the second one is calculating the icann victor's which are going to be related with This matrix basically becomes a P response to the wagon factors. So this is the procedure for this exercise. So first let's calculate the easiest and that is the first step diegan values. Okay, so to calculate taking values, we need to consider the the determinant of the following matrix a minus lamb that and identity matrix in this case is the determinant of one minus lambda 0001 minus lambda. 0011 Uh huh. Okay. So this determinant is equal to from the cube -3. The Square Plus 2 λ.. And this determinant should be equal to zero. So that means that we need to equate this polynomial here. That corresponds to the characteristic polynomial of this matrix. You will find that infant more advanced books you can find with dissertation triangle key corresponds to the characteristic point annual. Associated to some uh matrix. Yeah. Great. So here we got this Anchorage is characteristic polynomial and we need to find the rich and the roots. In this case you can factories this and you will think Lunda that multiplied to land the square minus three lambda. Lost to. So from this we got a radio solution that is lambda Equals to zero and the other one. The other solutions are lambda, he goes to two and lambda equals to one from this part. Okay, so these are the values for these matrix, so let's put them in descending order. So lambda one is equal to lump two is equal to one and the third I can I can I can value equals to zero. Now with this we need to calculate the We're going to the 7th step that responsible calculating the Eigen vectors. And for that we need to go to consider each wagon value. So let's start with the first one. And that means that if lambda is equal to two then we got this system, we need to find a solution for this system. So in this case I'm going to regret here de metrics a Okay. And yeah, So if λ is equal to two then this system becomes -100 minus one. And here X one X 23 Equals to 00. We can reduce this matrix deviation form and we will obtain here -1000 -11 000 Okay, So we need to find a solution for the system and that is no more than having this going to the green. Mhm. So the solution for the system Is that X one is equal to zero And that X two is equal to X three. Okay, so based on this, the Eigen factor and here you cannot do that. We got a free variable. Also you can see that on the invasion metrics here we got that these come corresponds to X three will be the free variable. And here we got two pilots. So we've got solutions for X one and next to So extra is a free variable and that means that we've got some constant value A X one is equal to zero and next to is just X 3 21 1. This here 011 corresponds to the iron factory. So for the one equals two to the associated Eigen vector is 011. Okay, so I'm the one is the constitution and the Eigen vector is zero. Right? What? Great. Now let's continue. Now we need to consider the egg in value two is equal to one and let's see what happens. So in this case I'm going to go in faster because explained the idea. We need to find the solution for this system. So a minus lumber too. So this part here is equal to The system is equals two 000001 and zero one. Yeah. Okay. And from this dissolution, is that in this case X two and X three should be equals to zero and X one is a free variable in this case. So X one can be can take any value and X two and X three need to be equal. So the Eigen vector Will be A Times 100. And this here is the Eigen vector associated to his identity. So for them the two equals to one. The wagon factor Is 100. Great. Okay. So λ one lump or two. And the associated Eigen vectors R 01, one and one year. Do we want we have one more Eigen value remaining. That is lambda Equals to zero. The case we return to the case A X equals we need to find a solution for this system Because I'm the is equal to zero and that is just finding the solution for the system And this reduced to one 000112. Okay, so we got this system and the solution in this case is equals two X one equals to zero X two equals two minus X three. And in this case x three's are free variable. So from this, the Eigen vector, this pan, these are convicted is equal to a 0 -11 And this is the 3rd iron vector 0 -1, 1. Great. So we got everything in place. So first I'm going to put it this way and I'm the one equals two to the wagon victory East 011 for them. The two equals to one and three equals to zero 0 -10. I'm sorry. It's my 0 -1 one. And this here is equal to 10 Yeah. Great. Okay, so what's going to be the diagonal matrix D. Are just the Eigen value. So it's 210. And the matrix B, as I mentioned, corresponds to put in the Eigen vectors um as the columns of this matrix. So that means here 011, -1 is important that you put the wagon vectors in order. So here we put the wagon value to. So here is the Eigen vector associated to that Eigen value. That's very important. Or what you're going to obtain different results. And what we need to check actually is that indeed the is equal to p inverse times A. Times. So just to uh go faster in this case the members of this matrix speed is equal to zero on how health. One zero zero & 0 -1. Hell one. Health. Okay, so let's compute So 1st a. p. This is equal to 010 200. Yes, 200 and then PM burst times 18. This is equal to so you're a one half, one half, one 000 minus one half one half times the matrix ap that we have conquered before zero,- 00- 00. And this is equal 22,000, And this is the the economic occurs.

Today we are going to solve a problem. Number 62. She had given three bedroom taxes, find food, find a full, then find 56 Kidded spying Fight 1.36 Then point a full seven. It's flying to fire. Nine. The little blind things too blind blinded. So it's inverse matrix will be one point of doubles, Right? This has to be heard. Calculated isn't currency. There are point full 966 minus 0.275 Mine is little pointed to 4 to 6. One point is it'll one play, then my nose little pointed to wait for minus their appointed 37 There, too. Minus 0.280 You are blinded 11 since thank you.


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