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'0 = X+ ,X9T + #Xb9 0} uo4nios IeJaua? a41 puiyJ...

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'0 = X+ ,X9T + #Xb9 0} uo4nios IeJaua? a41 puiyJ

'0 = X+ ,X9T + #Xb9 0} uo4nios IeJaua? a41 puiyJ



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$$7(\cos 0+i \sin 0)$$

Get so here we have switched room with variable were using, but it's all the same. We have ex prime uh, Plus signed here X is equal to zero. This means R p of t is equal to sign of tea and this does not have any, um is to find everywhere. So there's no the single point. So are above equation will have a solution of the form exit T is equal to he some from zero to infinity. End times t to the power, then, okay. And the power Siri's expansion of science t is equal to he minus teak. You'd her three factorial plus t to the fifth over. Five factorial minus t to the 7/7 factorial and so on. So that switches signs as the odd exponents that show up here. Okay, so we can go ahead and rewrite. Take what we can take the derivative of this except t So your ex prime of tea, it's gonna be equal TV some. And when I start at once, this is this first derivative, the end times eight and a T to the power of and minus one. Okay, and then we can go ahead and use that, plus the above two, um, equations to rewrite our initial expression here. Right. So we're just going to write this in terms of Ah, this first one will replace X. The second one, we put a sign of tea and the third one we replace Ex prime. And so if we do that rewrite, we'll get the defense Some from one to infinity End times a n t. To the power of and minus one plus the most thing for sine of t t minus teak cubed over three factorial. And so on times that some from zero to infinity n times teach the end and we set that equal to zero Okay, so we can expand the Siri's And what we would end up getting is expand the first Siri's and we get It's a one first to he played right, and but to a to t plus three a three t squared and so on. And then plus, and then we have this t minus t cute of her three factorial post t Feiler five factorial. And so on times. And then this other. So much is this could be a not crust. They won T plus hey, to t squared so on. And that's gonna be zero. Okay, so there's gonna be some, like, terms in here. It's going to multiply the second one out on and take the common coefficients and we end up with is a one plus to a to T plus three a three t squared and so unbelieving at that, too. And not t plus a one t squared, plus a to t cubed. Yes, this is because we're taking this whole thing times t that, Teoh Negative. Hey, not t cute over three factorial minus a one over t four three sectorial minus a to times T five over three factorial and so on. And we'll do one more here So that Teoh a not times t 5/5 factorial plus a one over t 6/5 factorial plus it to 37 divided by five factorial and so on. Okay, just keep following that forever and Saturday equals zero. But now we can combine our t terms, right? And if we do that sense of being equal to a one plus to a to plus a not Times T plus three a three plus a one times t squared Plus for a four plus eight to minus a not over three factorial times t cubed. Stop there. Okay. And so we want to set all these coefficients equals zero. Right, sir, this in prize a one must be equal to zero case. We get to a to plus a not must be equal zero. This says that a two must be equal to negative A Not over to. He said I'm going to the third one here. Yeah, says that three a three plus a one must be equal zero, but a one is equal to zero. So that means a three. It's gonna be equal to zero. You like zero mix things easier usually. Okay. And then for this last one that we need, we'll do 484 plus eight to minus in over three factorial equals zero. Um, but we can, of course, substitute a two for a negative and not over to. And we can no three factorial. We can sort that out. Sets of questions three times. Two times 16 Okay, so we end up with is a four is equal to a not over six. Usual arithmetic. Ter so all that out? Uh, okay. And so that we can for linen and I So our general solution waas Except he is equal to the sum t equals zero uh, A and TN, which of course, is just a not plus hey won t plus a to t square and so on. So we seven our initial condition which waas but x zero is equal one Okay, so that means are a not is equal to one. And since we have a not we, then that that a two's gonna be negative one half a four will be 16 and we can plug a lot in So we get our final solution is able to a non which is one, um, first, a one times t but one is could be native one house would get So get a one times T Anyone was zero. So we skipped that one and then we get plus a two times t squared and are ready to Was this native one half here. So you negative t squared over two, right and then are a three was equal to zero, but are a fours 16 So that's gonna give us to the power of 4/6 and so on. You can do a few more times if you like, but this is just a

Hello. We have all the number five questions is solve the question. Okay we have to find the Halifax basically determinant is X. Plus A. X. X. X. X. Plus A. X. X. X. X. Plus a. Equal to zero. So let us just ah operate are one are one place are to place are free so this will become Explosive Explosive. three x plus A. three x plus a. three x plus a. X. X. Plus A. X. X. X. Express A. Will be able to zero. Now let's take three explosive common. We'll be left with 111 X. X. Plus A. X. X. X. X. plus a equal to zero. Okay now we have to operate to get uh get 1002-0 here. So we'll be operating C. Two C. To minus C. One and C. Three C. 3 -1 1. Okay So three x bless a Si Tu -7. So C. One will be as it is one xx zero A zero 00 Okay And it should be equal to zero. Okay? It implies three express a. Into that is operate like this. It's quite equal to zero. So three x plus equal to zero X equal to -3 x three is the only solution. Thank you so much.

This video, we're gonna go for the answer to question number 29 from chapter nine point. So we have to find the values of our for which the turbulent off a minus our eyes equal to zero where he was given by this by three Matrix. What? Okay, professor, we want to find Ah, yeah, the servant off any minus I So you gonna have are in the top left. Zero they were. We have zero, then one minus are in the middle. A easy at the element of a is one and zero and one zero one minus. Huh? The seven of that is gonna be well, evaluation it along. The top row going are tied by the bomb. Right hand at seven in the bottom. Right. Hunty biting matrix. Which is what minus r 001 minus R. This is just gonna be easy. Calculated is ah times well minus r squared. We're looking for the values of off which that is equal to zero, which are clearly just the values, uh, is equal to zero. Ah, What

We want together from mental metrics of the system. Exhibition 50 equals the metrics A on tabloid by X 50 we have the metrics e 0100 minus one 000 00 01 00 minus 10 And to get the Eigen values and the Eggen vicars of these metrics we subtract are from the magnet of these metrics. Why in us all minus R minus R minus R and you quit this new metrics toe zero that determine of these metrics we equated with you. Now we can get the value of our by solving this equation. First we start by minus or but the buoyed by the smaller metrics minus R 00 zero minus our one zero minus one minus are minus one mile tabloid boy. The smaller metrics we exclude the role and the column off the Element one. Then we have minus 100 zero minus our one zero minus one. Finest are and these equals zero. The first term simple voice to be minus are month employed by minus are about the blind boy minus are multiplied by minus art which is R squared minus minus one gives plus one minus minus one plus. But the boy boy minus are minus. R equals R square minus minus one gives a plus one. You notice we have a common factor. We take this common factor out R squared plus one. But the blade boy we have remaining from the first time we have our square, and the second thing we have plus one, equals zero. Which means we have r equals posted or minus high. And this is a repeated I again. Very. Then we can get all for equal zero because we don't have a real bark off this wagon value. And we have later equals one, which is a coefficient, the imaginary boat. And the second step is to get the Eigen vector corresponding to these Eigen values. We can put our equals I in this metrics, and we can get the following. We can get that. Why in this, Ali one 00 minus one minus Ali zeros You zero zero minus. I won 00 minus one minus Ali. And these metrics is month of Lloyd Boy. The wagon victor is that 123 ends at four. And this multiplication is equal to zero. By getting the four equations off this multiplication, we can get the again Victor. The first equation is minus pi deployed by said one plus two equals zero. The second equation his miner's that one minus, I said two equals zoo. If we noticed that these two equations are the same, which means we can let said one equals any value, for example, is and get zipped into interment service. We can get the two equals from any of the questions they will give. They will give the same answer they do equals always. Similarly, for the 3rd and 4th equation, we have minus oy said the three Less than before equals you and minus said the three minus always at four equals zero. And these two equations are also the same. Which means we can let the three by any value, for example, V and we can get that four in terms of B, which is already Now we have the again Victor that equals this all. Yes, the ivy. And we know that we have here toe again Victors. But we have won only one Eigen value. We have used only one Eigen value, and this is the property off their beated Eigen value. We can get a generalized again, Victor. From this exhibition, we can get to again Victors by booting First weekend Put this equals one and the equals zero. And then we put vehicle zero and we would is equal zero and vehicles were by putting its equals one on vehicle zero. We can get the first again. Victor. They don't equals one I 00 and we can separate the real board and complex work or the mission abort 100 zoo plus 010 The blood by I. And we can do the same by letting V equals want. And it's equals you. We can get that two equals 00 one i and it equals 0010 plus 00 01 month tabloid boy. Now we have a and B for the for both. Like in Victor's. We have a The inspector is eight. This victories be we have year A and be for the second again, Victor. Now we move on to get the fundamental metrics from knowing a and B for both the again victors. But we have to do that. We have a generalized victory which means we employ in this formula to get the solution x one of t equals e to the bar or won t when the blind by the victor You unless the tabloid boy the metrics e minus r I month employed by you where you is Eigen Victor, we can expressing we can This this equation for any Eigen value, for example, you start early, you are We can put our equals Then we have e was about off duty. But the boy boy the again Victor You can use any of the Eigen vectors. We have one 00 for example, plus t But the blade by the metrics a minus are which is minus oy one you and zero The breakfast metrics minus one minus Ali 00 00 minus one 00 minus one minus. I not employed by the victor you, which is we have used you equals 10 and zero one. Then Holly zero owns you. And we can look this up this term. This multiplication gives us a zero victor. Which simple voice to be thes. Who there miss? Only And we can. As usual, he was Oilers formula to get signed on because I off e to the power of I t. But we have the chick and do be sure that disturb this third cancels out the other again. Victor, we put our equals by and used you. Acquitted two which waas 001 as we remember then we test x two of tea equals you to the about off our t its remote of authority. But the boy But you we know. Have you 001 i t but the boy boy the metrics a minus, Roy which is the same metrics? Minus one minus all you want. 00 minus one minus their 000 minus I 10 minus one minus. And we multiply it by the victor. You, which is you 01 You can notice that here also this multiplication give zero. We have the first two. The first two rows is 00 The third row is minus. I plus I minus I loss. I it's just you. Uh, the last term is minus one minus y squared, which is minus one plus one equals you. This means that the's third cancels out and equal stools. You and we have only this bark off the answer, which is the normal answer. We know now we can get the fundamental metrics by getting the value of X one of t. We know that it equals C one tabloid boy. He was a lot of Alpha t, which is zero because we have the roots are equals first of minus all we have all for equal zero and beta equals one. As we said before, it is a lot of all 30 design. May 30 multiplied by the victory which was 10 you zero minus years of our Al Fatih was I in science 30 sign May 30. But the blood by the Victor B, which was 0100 and extra 50 equals C two. But tabloid boy eat about of you sign 1000 plus e is a lot of zero cause I t multiplied by 0100 and we can continue to with X is 3 50 using the other day and be with we he was Here's another A and another B And here's another day and another be toe get extra three and export of tea. Now we can write the fundamental metrics. Ex softy to be the first victor. It comes from Ekstrom. We have design T. They have minus e. We have it was a volunteer is one. Then we have minus sign t 00 The second victim is sign a design t zero and zero. And by getting X three and X for which will be the same this we will have the same board because it's identical. We have year 0000 cause I ain t sign d. My nurse I inti co sign t and this is the final answer of our problem.


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