5

0I 1 U 1 { 1 1 388 1 1 1 1 1 L 1 3 1...

Question

0I 1 U 1 { 1 1 388 1 1 1 1 1 L 1 3 1

0I 1 U 1 { 1 1 388 1 1 1 1 1 L 1 3 1



Answers

$\left[ \begin{array}{lll}{1} & {1} & {1} \\ {1} & {2} & {3} \\ {0} & {1} & {1}\end{array}\right]$

This problem will be to determine explanation regions each of power a. T we're a excuse me tricks Cubans here. So there's so much of this problem is done in two main steps. The first step is to find a generalized idea in Victor's Off the matrix A to determine the fundamental matrix X t And then we will use this formula here to compute explanation. So let's start. So we need to find first the eye gym bodies I can values are given by find First off A, which is the determine off the matrix a minus R types that entity which to return of rejects one are one, 21 sergeant one term in office matrix is keeping bythe are three plus three It's for these factors Hi Directive Our plus one times our minds choose So then we need to make this equal to serial five sports That first solution given by this bracket people zero so are equal to its form. Second solutions conveyed this bracket bracket, but zero so are you. So we also need to take into account the multiplicity off each item about this bracket. Here are my our boss one has exported while they're here, they're placing the Fuze I am. Value is just one on Pierce ones in these factories. On the other hand, the idea of value are equal to chew. Appears twice so therefore, commit complicity off these. I hear you. So we need to take this into a calm to find the general ice age in gloves off the rejects. So let's find the IJ in Victor's after Richard's A. So the first line for I think most pace off inmate traits A plus, I So you see, for our equal that this matrix is the major scale to one 2 to 1. Cyril. Okay. Okay, so this is quite willing to dominatrix one Cyril cereal 12 Sure. So there's no space of this matrix is generated by the victory you have won. That is 34 So this is our first general. It's a joke. So remember that they want to place a t of thes ie in by you. So the molars pays needs to be one in this case, but these does what? Okay. On the other hand, look at the next item value are equal to you and a So this is the matrix. You got four children. Syria. Individual. So the competition of the noticed face off these past that nature most peace you can do you If you want to compete yourself, he's hopeless. Okay, so remember the multiplicity of this I Wow. So this knowledge space is not a generalized I thank you off our eye about because you mentioned a mention of the No space is not the same husband off 10 minutes. Finding more space off the matrix. A minus. True. I escort this squired power here is the same as the multiplicity. Both our idea. Thes magics, squirt. He's the maid. Breaks given, then by great tricks. Three big tree tree four for four. It is a cooling to the matrix one production and this nose based bus half two generators. So the 1st 1 he's Cyril sickle one three. So they tell you why from the agent are these? I don't think those are two children. Allies ij inspectors to that charming and matrix ext. Okay, so the matrix X t is the hatred skin, then treacle x one T x to t x three. Okay, so this these columns each of these columns come from eats generalize idea. So let's find first, then x one of t that comes from the agent already before. And this eye you, Dexter. Negative tree for thes kahlo is giving thing but into the power teat Times tree for so things to make tea here comes from this Let's find the next trip so x too cheap com agente are equal to do you, Doctor, You're Jewish one Ciro. So remember, this is the inspector come from the normal space with the matrix A minus two. Okay, so he's not a regular i ice agent Victor that comes from these non space in this case. Then this'll calm will be the matrix, given by each to tee times plus team times a true t they think a wings to I. So we need to find them. The implication of this matrix thing tax you choose. These terms depended. So this will be easy to t means defector. Why, Cyril one Which the Victor Jew Cyril Ciro T h t The identification of this matrix times He's the victim's shoe. So we get us a result. Yeah, e t he to achieve waas t t. All right, so let's do then X three seem lurched off. One. This will be eating the power to t the victory three plus tee times to t A minus. True, I That's your trees, old sergeant. Allies No. One case. This would be to tea things once. One times it's your wife. Why? And again, this is each report to G. T. G. So we can write down our from the mental tricks. Asked the magics x t quite cheap tricks. G g Cheers for tea achieved. Plus, she keeps calling to cheer t each team to poet e T minus the h g way Need to compute the inverse off this matrix Emma Lloyd a t e contest. So wait, this magics reaches found t e quanto 00 we get in matrix one for you in the Matrix for the Regis four. It's in birth matrix off fighters. That, then, is to multiply his big things here on the embers. When we do these with the kitchen's right there, whole missus, your final answer will be three h t that suits you team first roll. It's plus for T six. You get for tea for five here lines for tea plus 13 Tiu here. Keep me too. 60 Jury team T team. I will be true. T I must take Tiu the fire.

So here we have a big Five by five Matrix, and luckily, we are going to use software to do the heavy lifting for us. Okay, so this is our given five by five matrix here. Okay? And so we put this into some sort of software that will find Eigen values, and we get that the Agon values are one 11 eye and negative. I okay. And then we put it into, Ah, another algorithm that will give us the vectors. And those are the first time was just one of the top and then zeros. Next one is just a one in the second place. The 3rd 1 is a one in the third place. It is a little bit excited. It's not gonna be an identity and tricks. There's an eye there. And then we also have, um, one and negative. I Okay, so these there are i n vectors. So, um, now we can also get the big x of t from this, and so it's going to be really big. Um, can it gross monster? The matrix here, when minus t plus one have xi squared. He's a t 1/2 xi squared. U t t plus 1/2 t squared either the tea 00 t minus t squared. He didn t one minus t minus t squared either. The tea plus two squared into the tea. Ah, here we would have negative three t minus t squared he to the t can t squared. I don't have t squared e to the t um t plus 1/2 a t squared. He didn t and one plus two t plus 1/2 a t squared times e to the t. Okay. And then we just have this Lessel corner to deal with. And so this is gonna be our co sign of tea. Scientist e negative sign of tea and co sign of tea. Okay. And so then we evaluate the, uh, the X zero. Really lovely thing happens. All their teas go to zeros are signs go to zero. And when you cook it out, we actually get the identity matrix said just ones across the diagonal. And that means ah, that the inverse is also get any matrix. So when we go to calculate, he'd, uh, big a t being equal to this exit. Tee times are in verse at zero since the M percent zero. Is it any matrix? It's just gonna be equal to Exit T. So we already figured that out. There is nothing left to do. And the answer is this big guy here.

Here we have the Matrix A is equal to It's a three month three matrix again. So negative 110 and then 1 to 1 03 native one. So the characteristic equation for this one where this matrix iss negative Lambda cute plus seven Lambda plus six equals zero seven lander for six people too. So we get that Lambda one common to collar three is equal to negative one negative too. And three now for Lambda equals three, we can solve the argon victory equation and we end up getting that diamond record issue equal, Steve times 1/3 4/3 and one. And for Lambda equals negative too. We get you is equal to a T times 1/3 negative 1/3 and one. And for Lambda equals negative one, we get u equals T types in native 10 and one. So the general solution except E can be written as C one times Eat that. Three tea three tea times 1/3 4 3rd 1 plus C. Two times e The native to T times 1/3 native 1/3 1 Thank you. Plus C three times e to the negative teeth comes native 101 So this is our final general station

Okay for this one. We have a is equal to 12 negative. One 011 and zero. Negative 11 So the characteristic equation for this problem is given by negative Lambda Cubed plus three Lambda squared minus four. Lambda plus two is equal to zero. And when we solve this equation, it's a cubic equation. So you will get the three argon values as 11 plus I and one minus. I noticed that these talking visor complex congregants. So now if we have the Lambda equals one, then upon solving the system, eh? Minus I times u equals zero. We will get that use equal t times 100 and for Lambda equals one. Plus I, using a similar process, we end up getting that you sequel to a T times I'm minus two negative I and one no, for Lambda Contra Kit equals one minus I, which is given here. Then that implies that you is equal to it. Turns out that the Egan vectors are also conflicts congregates of each other. So you was simply gonna be equal to a T times negative. I'm minus two guy and one


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