5

1 1111a...

Question

1 1111a

1 1 1 1 1 a



Answers

$A=\left[ \begin{array}{rr}{-1} & {1} \\ {a} & {-1}\end{array}\right]$

So we want to use Aiken values, uh, to find the stability properties from for the equal of Rhea for the following matrix. And we want to look at all values of its stay and see what happens. The first let's find the Eiken values. So to find the idea values we take the determinant of a minus. Lambda I So we have negative one minus lambda and one mine. Excellent. So the determinant is equal to negative one. Linus Lambda Times one minus lambda. Um, minus no product of the off diagonals. So we have minus a and we want to see when this equal zero eso distributing everything out, we get negative one plus lambda minus Lando plus Lambda squared minus a equals zero. So, plus Linda minus land, they can't pull out and we're left with Lambda Squared equals a close one. So Lando wanted to or equals who? Plus or minus squared of a plus one. These are our two Aiken values. So now let's look at, um, our equilibrium for the stability. So, uh, just just a reminder our Linda are equal to plus or minus square to April's one. So first case is a equals negative one. So what happens in this case? Um, lamb don't want him to are both equal to zero because we have plus or minus zero. So we have, um, infinitely many solutions since the determined of a would be zero infinitely many selections Case to, um, a is less than naked of one. So if a smaller the negative one, um, the negatives gonna dominate. So we have pure imagine Aires Aiken values. So this just means they have no real part. Um, because they would be equal to square root of, uh, negative April's. Won't I Soap your imaginary Aiken values tell us that we have a center that is our equilibrium and last case. What if we have rely on values So if a is greater than negative one, so positive dominates. So we just have plus or minus squared April's once, um, and that would be really so if we have to re Elikann values, Um, we have opposite signs because you're a plus and minus. So this tells us we have a saddle point, so we can either have a saddle of center or infinitely many solutions

The question here gives us a matrix is equal toe a 11 a. And it wants us to solve for the Eiken value. So we have to plug this into the characteristic polynomial X squared minus a plus D X plus 80 minus BC. So if we plug it in, we get zero is equal to X squared minus to a about my X plus, um, a squared minus one. So essentially X squared minus 28 acts. Um, plus a squared minus one is equal to zero. So when we saw for this, we get the values. Of course, if we won, we solve it. With respect to the quadratic formula, we get the Eiken values of non the ones you go to. A possible one, and landed too is equal to eight minus one. And that should be our solutions to our question here.

We're given this magic A We're universe first. Me from the convertible meeting room. Without it, the determinant they You could have zero in a but not in veritable works. Check it. A convertible. What do you say? Actually, I read it down here. Let's find a determining a check of the convertible or not. 111 First, I'm gonna high road to buy world one by negative one and had it wrote to So I guess one might one minus one zero. Making one plus 21 No one here. Next I'm gonna multiply growth three by negative one times wrote to I get 111 negative. Q one is negative one. You know, the determining the mortification old the numbers in the pivot in the diagonal interment is clearly not so. Therefore, we can find a neighbor nullifying chambers through the over inside the right chambers on this side. Very eight in the side. You're a here you have the identity matrix for three by three One here is you here alone? Now we're gonna really do until this side here. It looks like this. Once we do that, we will get a members on this side Look for reduced. Well, we already thought before we're finding the determinant. Do it again. First they can about this here. 11 now weaken Can't hold this position here. Negative one. They won negative times. Negative. 101 Negative. 101 Here one. Now weaken. We can scale the throw here. We can divide the group by negative one. We get negative here. Also here. Positive here. Now I can scale road three by minus a few. Added row to cancel this The negative too. Times road Here. That positive you minus one. That's one. Make a few plus one minus one. Two zeros too. You get one Next. We just need get rid of this. We can. He gave the period. Rowing added to the first would be a bit of this one. Here. You never get here from zero and in one one to negative one plus 01 and 101 Now you get a second road out of the first road in negative. One plus two. Just one. You have one plus minus +10 You have minus to plus one. You hear? This is the identity matrix implies on this side. He had a members say in verse, should be one bureau minus one one, minus 12 and minus 11 minus 11

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.


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