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Problem 3_(40 points) The matrix13 ~14 12 17 ~24 25 ~14 16has two distinct eigenvalues with A, < AzThe smaller eigenvalue A,has multiplicityand the dimension of ...

Question

Problem 3_(40 points) The matrix13 ~14 12 17 ~24 25 ~14 16has two distinct eigenvalues with A, < AzThe smaller eigenvalue A,has multiplicityand the dimension of the corresponding eigenspaceThe larger eigenvalue Azhas multiplicityand the dimension of the corresponding eigenspace isIs the matrix C diagonalizable? chooseNote. You can earn partial credit on Inis problem_

Problem 3_ (40 points) The matrix 13 ~14 12 17 ~24 25 ~14 16 has two distinct eigenvalues with A, < Az The smaller eigenvalue A, has multiplicity and the dimension of the corresponding eigenspace The larger eigenvalue Az has multiplicity and the dimension of the corresponding eigenspace is Is the matrix C diagonalizable? choose Note. You can earn partial credit on Inis problem_



Answers

Use some form of technology to determine the eigenvalues and a basis for each eigenspace of the given matrix. Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$\diamond A=\left[\begin{array}{rrr} 25 & -6 & 12 \\ 11 & 0 & 6 \\ -44 & 12 & -21 \end{array}\right]$$

Hello there. So in this exercise we have some statement we have some situations that represent some transformation in our three. Okay. And we need for to each operation transformation we have an associated matrix. We need to find the wagon values and taking pictures for those matrices. So let's start with the first one. The first one is a reflection about the xy plane. So the matrix is represented by this 100010 and 00 one. So you can obtain here the agony vectors and Meghan values using just algebra at the point of this exercise to do it dramatically. So here we have this space, Y X A Z. And he will he will have negative values of the negative values of X. And here negative values of Y. Where is going to be defined the eggs waypoint. Well, are all these points here. Right. These represent the xy plane. Now what's going to do this? This reflection is taking any point in the space for example, here be a zero and after applying the transformation we past phony or finally through these plain and reflected. Another position here. A easier we're going to say that this transformation is called a Okay, so that's idea here, were we are reflecting with respect to this plane? No. What is going to be the Eigen vectors of these metrics? The egg in space? Well, the again, space are going to be defined by the vectors that maintain environment. Right, So the environment space and you can observe clearly the xy plane is going to be in variant under this transformation because basically if you pick some point here in the plane you're going to end in the same point or in other words a let's call this point alpha, He alpha returns you. Okay, so what's going to be that? Those vectors, what's going to be the wagon space? So this space that's going to be in variant under this transformation as I mentioned, is the xy plane. But let's find a basis for the X. Y plane. And that's really easy. The vectors the basis vectors Are going to be the one. Once good, It's going to be the vector 100. And the second picture will be the vector 010. As simple as that. Because with those two vectors that is one aligned with X and the other one A line up with Y basically only find B one and B. Two. We can spend the whole X Y point. What are going to be the Eigen values? Well, you can observe that if we pick this point we're going to return to this point again at this point we return to this point. So basically we have just one egg in value that is equal to one. Let's continue. Now we have the final projection onto the xz plane. Why is he playing how this Will this matrix? A This transformation looks like this 011 and the rest of the matrix are going to be zeroes. Okay, okay, so we have this matrix again, the geometric interpretation of this orthogonal projection onto the Y. Z plane is the finest fall. So here we have the Y axis, the X axis and easy access. They got their values and they've got the values in here. What is located the Y. Z plane? Is this point here? Okay, so this whole plane hunt infinity is the Z. Y. Place. Now what is the meaning of a projection on to this point? Basically if you pick some point, let's say over here. Okay, get it. And this place. Okay, so this vector let's put let's take this vector B0. And then after applying the transformation, what you're going to do is projecting onto this point, basically you put it or throw a new author finally come to the why is the plane? So that's going to be the point. A busier. Okay, so now what is going to be the egg and space of this transformation? What's going to be the point that our environment under this transformation? Well, I think that is clear. Is the Z. Why the why is he playing basically? If you pick some point here, if you projected or finally onto this point, you're going to obtain the same points at this point here. Let's say this. Mhm sigh. If you apply a to slide obtained at the same point. Okay, so no change exciting is the caustic sign. How to generate this space. What we can find a basis. Let's just pick the this is the one, the unitary factor in the Y. Direction and the unitary vector in the direction. That means this plane can be spanned by the vector 01 zero and the other vector 001. And you can observe that for these two Eigen vectors that we have here, we have one Eigen value. That's going to be one. Towanda equals two. There is another pointed, his environment and this transformation clearly the origin. But that's the trip. Let's continue. Now we have three the counterclockwise rotation about the Y axis. Okay, this matrix look like this to do the tradition matrix coastline of theta zero line of feta 010. And here minus line ft zero and coastline. Okay so this is generic rotation in the Y axis. But we are interested in this case, In those rotations of 180°. Okay so how look this rotation when you your first irritations about the Y axis means that if you consider here indeed my ex the space coordinates. Okay and you pick some point let's say over here you can rotate this in a cone. Okay, all these points are going to be available for irritation. And this cone here. So there's a point after some this rotation eight. You are going to by in some point in the circle and the axis of this cone of this rotation car. Well it's going to be the Y axis. So that's why it's about the so clearly here, the on the well here we have a rotation of 80°. So that means that we take this point be here and we put we transpose we we moved to the other to the opposite side here or a week. If you are located here. Let's see here. Let's see, sigh. Then you take a 180° rotation. That means to the opposite side of the circle and you're in here. Okay, sorry. And in this case is going to be a P after applying these transfers. Okay, so well you can Put here the 180° you will obtain a different matrix after evolution. Okay, 80 degrees. This is going to be -1. There's going to be zero. There's going to be also zero. And this will be So this is the final matrix of these kind of rotations. Okay, so what are going to be the point that are invariants under these rotations? Well, clearly the access. Right the y axis, it's going to be in variant under dissertation. If you pick some point here in the UAE because the rotation about the y axis, you're going to take the same point basically. So if you pick this point let's say um all for then a alpha is going to be the same point here. So do you need to your iphone space is going to be aligned. Just the the fine by the way access So your icon space is the Y axis. So a picture basis for that. It's going to be the vector 010. And that's your only Eigen vector. And we wanted this victory maintain environment basically if you pick this vector, your turn after playing this this transformation, it returns you the same vector. So a. B1 is supposed to be one. No risk killing of the picture. That means that lambda, it's going to be white. Okay because these are you have multiple water. Finally you have the violation. The dilation is the simplest transformation that you will find in the space. Why? Because it's just the identity multiply by some scalar at the point for this kind of transformations. Is that the matter which vector you apply to this transformation, you will obtain the violation fracture time, the same victor. So basically it's like kind of saying that all the vectors here in our three Are the all the vectors in R three r the Eigen vectors. So you need this your dragon space. In other words, the earth with the whole space is your big in space. So you need to find out a basis for this. So your basis for your wagon back for your dragon space. It's going to be you can pick any any any basis for our three. So I'm going to speak the the economical one. So 100, 010 and 001. These are going to be your Eigen vectors, the basis for your own space. And here you're, Eigen values are going to be just K.

So for us to die analyze this matrix. Since they don't give us the Eigen values, we need to find those first. So remember to find that we look at the determinant of a minus. Lambda I and so a minus lambda I is going to be so let's just write that So it would be determinant off eso for my slammed for my slammed to mind Slammed to my slammed We have a one in the corner here and then everything else is going to these zeros. And luckily for us, since we have it in this upper Earl Starr, this lower triangular form, we know that the determinant of this is just going to be equal to the diagonal multiplied together. What? I don't know why I have a zero down here, but that zero should be, uh so this ends up becoming for or minus slammed squared times to mine. Islam two squared. And remember, this should be equal to zero since if we were actually thinking about what this is. We have a minus slammed I, Times X and this being able to the zero vector. So this thing here should be zero. Because, remember again we're assuming that X is not zero. Come on now, since the determinant of that is going to be zero that just tells Islam dais for land is too. So we'll use these values here to help us create our factors. So let's go ahead. Pull this down. Let's first check out when Lambda is equal to, um to actual. Let's do four first. So a minus slammed. I, in this case, is going to be so 0000 very sparse matrix and sparse just means a bunch of zeros. Uh, that one stays there and then negative two. Okay, so we have this. And now, um, we'll go ahead and really reduce this, which I actually went ahead and already did before and doing this e mean, we actually essentially just get what we have here. So I actually don't even need to really do anything, because if we look at it from here, this is just telling us X one whiteness to X or is equal to zero, and then we have X or negative two x three is equal to zero. So first this implies X three is just equal zero. And then this top one here implies X one is equal to two x four and then notice we don't have anything about X two. So x two is just some free variable. So now let's go ahead and set up our actual factor, our Eigen vector. So we're going to say X is equal to so it's x one x two x three x four So x one over here we said was going to be two x four x two was just free. Nothing really There x three we said was zero and then x four is also free. So it's just going to be itself. So maybe here I should say X two x four, both free and now we can break this up. So we have X two here. Eso we only have a next two in row two. It'll be 0100 plus x four times. So we have to and row 11 in row to none in a row. Three. I'm sorry. Uh, there's none in row two. That was to their I mean, that was executed on down here. We just have one. So remember these are going to be our to Eigen vectors. or Lambda equal to five right now, Not be able to five family able to four lambda you before. Now we're going to repeat the same process, but with two this time. So let's sweep this down. So now when Lambda is equal to two, we have a minus two I, which is going to give us 20000200 And then everything else becomes zero. Except for this one in the corner. Um, and then this here. Well, the one in the corner down there, just as Excellency zero. So we could just kind of ignore that. Um So what this says, though, is two x one is equal to zero. The second one is two. X two is equal to zero. And then the last one just says X one is equal to zero. So not just implies X 100 x two is equal to zero. And then we have x three on explore being are free variables. So once again, we go and set up that specter So x one x two x three x four. So we had x one x two book being zero and then x three x four just themselves. So if we split this up just like we did before, we only have an X three in a row. 30010 and then x four we only have in row four. So 0001 and then both of these go with Lambda is equal to two. So let's go ahead and set up our diagonal eyes and matrix now So we have d is equal to um So I'll do 2 to 44 and then everything else is going to be zeros. And now it doesn't get better if you do 2 to 44 or 4422 a. Za, Long as you have matched up the Eigen value with the Eigen vector which we do in a second. But this is just, I would say the most standard way to do it from smallest to largest now to get P. So we need to place a Eigen vector in the first two columns here and actually let me see how I set this up earlier. Um, I don't need to necessarily do it the same way, but I just want to do it so we get the same answers that I wrote down. So we don't have to do this all by hand again. Um, since everything already plugged into my calculator Um yeah, so you could see here. Eigen, Vector one Eigen Vector to Eigen. Vector one, Eigen Vector to And you could also flip these It doesn't really matter. You just get a slightly different in verse now for four s. Let's go back up here and look at what those were. So that was 0100 and 2001 Um, so let me see which order I had placed those in. So I did 0100 And then I did the other one 2001 Yeah, so there's gonna B P now we need Thio. Get P in verse. You can do this pretty much any way you want, but my favorite way is to just set up the augmented matrix and reproduce it into the are set up in augmented matrix, where this is equal to the identity and then just roll reduce it. So I'm gonna put P over here. So, 00020010 10000101 And then over here, I put the identity matrix, and then we're going to ro reduce this. And I went ahead into this beforehand, and that gives us the matrix. Zero 010 negative. One half 001 01 00 and then one half 000 So, uh, sorry. This is not exactly what it was because this should be the right side of the matrix scandal. Head of myself. So the left side, Woodrow reduce down to the identity. Yeah, and then the right side will roll reduce down to our inverse. So this is P universe. So let's go ahead and write out our diagonal ization of you So there's gonna be a is equal to so p d. P in verse. And so p we had was 0002 00101000 0101 and then d we had so 2 to 44 along the diagonals and then everything else zeros and then are inverse that we got was 0010 negative one half 001 0100 one Half 000 and so this is going to be our diagonal ization. And like I was saying earlier, you might have got something slightly different from what I got here. But it may be still a valid way to write the diagonal ization. As long as you multiply everything together and it gives you the original matrix say it doesn't really matter, because again, this is not a unique thing. We will have multiple different ways that we can write any diagonal ization.

Okay, so we want to die. I generalize this matrix a ah, and to give it the pagan values are given. So we have Randall want is one number two is to and then the three is through. So let's start Dagon izing. So what are we going to do? We're gonna find De wagon Victors, because since we have the egg and values when we want a equals p d p minus one, we already know the values off de since we have de again values. So let's find the egg and vector so we can fill p how to find the AIG in Victor's will for the first Hagen value of them, they equals one. We we try to solve this linear system so a minus identity times a vector X equals zero. So zero here's a vector for zero and X will be the a convict Age and victor were looking for one of the again vectors were looking for. This will give us assistant within our system that looks like this minus 24 to minus two minus 330 minus 312 times a vector x one x two x three from the components X and which is equal to a vector full of zeros. Well, that will give us three Viniar equation. So if you want to solve using matrix or in your equations, it's up to you as long as you solve it properly and that will give us the following system. So minus two x one plus four x two minus two x three equals zero minus three x Want Last three x two equals zero and minus three x one plus x two first two x three equals zero. If we look at the second equation here, that means that X one and X two are equal, so excellent he caused x two. If we inject that into the 1st and 2nd equation, we will get minus two x two plus four x two minus two x Treece of two x two minus two x three equals zero, and we'll get the same thing. Something very similar here except here. The sun will be opposed to minus three x two plus X two equals minus two x two. Thus, a two x three equals zero and, well, that basically means that there's no need for debt. You hear that just means, uh, let's start on in your name that x one and x two and x three are all equal. So x will look something like x one x equals x one time 111 So every component is equal and we're gonna use 111 as the basis forties dragon space. So I'm going to circle it and Green X one will be the first taken vector off. Well, they the first basis for Reagan space of a and it will be the first column of the major Expedia Natural. Now we continue for Lambda equals two. What party? Eggen vectors. So we solve it in the near system a minus two times and into tee times extra. We self that linear system. This will give us, uh, this new system minus 34 minus two minus 3 to 0 minus three one juan times x one x two x three For every component of Erector X, she is equal to a vector full of zero. It can be written as a linear system personally, a prefer to solve in your system for smaller dimensions. So for dimension two or three, I think it's always morning easy to visualize system that maitresse ease Trust foreks to minus two x three equals hero minus three X one plus two ext. Who cause zero and minus three x one person X two last X three equals zero. Well, and what are we gonna do here? Ah, we're gonna get sub strapped some light. So first of all, we that we have this second line here's a minus three x one plus towards two equals zero that give us that two X two equals three x want. And here we have four x two So four x two will be six x one So we will have two x two x two minus two x three equals zero and ah, here we have minus three x one. So two X two Shall we have, uh, minus X two plus X three equals zero. So here we just replaced minus three. Excellent. By minus two up to. And here you have it. So last linear system looks like this. So x two equals x three two X two equals three x one and again the last line will give us the same thing. So x two equals x three Ah, which is the same as saying that two x two recourse to x three two x two equals three x one and two x two equals two x three So the 1st and 3rd line are the same. Which means that the basis for his against space will look something like this so x two times, uh, to three through. So So this is the basis for his second against pay, which would Which is gonna be the second column of Matrix P And we can just senator process for a Lambda equals three. So we have another in their system a minus three times identity times X equals zero. Now you should be a bit familiar, so I'm going to skip right away today. The near system minus four x one my plus four x two minus two x three call zero minus three x one plus X two equals zero and minus three x one plus X two equals zero. So the 2nd and 3rd line are the same, so we don't need to rewrite them, which is great. There's already a lot of writing here, So we have minus four x one crush and here we're gonna use this Second entered lying to write X two equals my equals three x one So the first time will be minus four x one plus four times three x one uh minus two x three equals zero. Uh, we still gonna have it two equals three x one and up here we're gonna have eight x one minus two x three equals hero and it would put everything in the most simple for possible means. X three equals four x Ron and X to course three x one Their floor x is well, look something like this 134 So this is the third basis for the third hike in space and the third call off p and just give us the following diagonal ization. So a cool The 1st 1st Eggen Victor, who found was 111 The second basis was 233 and then 13 four for the last one. Find it 38 and values were 12 threes. We put them on the diagonal matric, Steve. And here we I won't computed and verse if you want. That's fine, but just writing it like this is a the most important part So here you have it. A in the form of P times diagonal matrix d, times P minus one. Pretty simple. We found the values Diegan Victor's based on the Hagen used implicating in the question.

Were destroyed. So. Yeah. Whereas your people previous exercise 9 14 for a new matrix T. So we use the three x 3 matrix with entries 3 -1 1 7 -5 1. I mean I'm sure And 6 -6 2. No shit in part A wants the first time. The characteristic polynomial of B. To do this. We need a few things trace of our three by three. Matrix B is three minus five plus two or zero. Be determined of our matrix B 78. Well this is going to be wow negative 30 -6 -42. Yeah the 70 73. We talked plus 30. The 73-7 plus 18 plus 14. So yeah visit old caprices. We got headlights. This is a -16. We can see where you have a bubble bubble caprices like it's it's kind And we want to find the co factors be 1 1 which is the determinant of negative 51 negative 62 which is negative four 73- two. This is the co factor of 3162. Uh huh. Those are the cars those cars like which is zero worth. And the co factor B 33 is the determinant. Uh 3. -1 7 -5. But like which is not great. Get into the car. Therefore it follows that the sum of our co factors B II is negative 12. No, We know quickly by three. Nature sees our characteristic polynomial. Delta is given by t cubed minus the trace 12 times T plus the sum of the co factors negative 12. I'm sorry CQ minus the trace zero times T squared Plus some of the co factors negative 12 times two minus the determinant to be. Now you're 16 plus 16 and he ran equal. This is a cubic polynomial factor. This well notice that if delta has rational roofs, then it follows by the rational root theorem that has to be of the form plus or minus one. Plus or minus to plus or minus four plus or minus eight Or across your -16. If you're lucky you'll test two. You can sometimes division. So are coefficients are 10 negative 12 16, two times 1 is two, two times two is four negative eight seems negative, eight is negative 16 and zero. We have a remainder of 02 is a root of our characteristic polynomial. And therefore we can write a characteristic polynomial as delta t equals t minus two times the remainder T squared Quotient I should say plus two T -8. Which we can factor the quadratic and we get t minus two times t minus two to minus two squared times t plus four. See Yeah, now the zeros Lender one equals 2 And landed two equals -4. These are the Aydin values of our matrix B. I'd like to see, I'd like to see it. Dude chuck. Now in part B. Just as in the previous exercise. Mm hmm. We want find a maximum set s of linearly independent Eigen vectors of the Okay, this is in the previous exercise, we're trying to find the basis for the Eigen space of each item value of the It's working in the first Eigen space lend the one equals two. Mhm. Like, well, we're gonna subtract two down the diagonal of be. So our matrix M is b minus two. I I don't know. And this gives us the three by three. Matrix one negative 11 seven negative seven suck one and 6 -6 zero. That's a that's right. And this corresponds to the homogeneous system, X minus Y plus Z equals zero, seven, X -7. Wine Prince Plus z equals zero And six X -6 by Equal zero. And this simple system simplifies to X minus Y plus Z. Our first equation equals zero. And then the second equation is simply Z equals zero. Only here it's a bit. So we already see that any solution has to have Z equals zero. So we only have one independent solution. For example, take X equal one in life. Then it follows that the doctor you With coordinates 1 10. This is a basis. Siebert For the idea in space of λ one equals 2. Right? Yeah, comments are funnier than me and most Yeah. Now consider the other icon space for lambda two equals negative four. So we subtract negative four down the diagnosed B. To obtain M. So M. Is the matrix B plus for I. This gives us the three x 3. Matrix 7 -1 1 7 -1 1 she's And 6 90 of 6.6. She kept coming around after he found open. Mm This corresponds to the homogeneous system. Seven X minus Y plus Z equals zero. Seven, X minus Y plus Z equals zero and six X minus six. Y plus six Z equals zero. This system reduces to the two equations X porn site. All right minus Y plus Z equals zero. You divide the third equation by six and You only like the fucker for like 30 seconds. six Y -6. Z equals zero. So you do this. I just most pain. Yes. This one looks like eliminated the 2nd and 3rd equation right through my body. Yeah. Yeah right possible. And we see that the system actually has only one independent solution. So take Z to be one. You see that? Why must be one And the X must be zero. So we get the solution Z. Which is 011. Send somebody down here. And since the only independent solution this forms a basis for the Eigen space of land of two, which is negative. four garden hose in his ass. Therefore it follows the process the set S with vectors U. And V. You just Or 110 And 0 1. 1 elegant. This this is a maximum set like shows both P. T. Barnum of linearly independent audience. I've been vectors for B. one of them was where the what is it? Finally in part C. Just as in the previous exercise we're asking he is diagonal, Izabal. And if so to find a matrix P. Such that the diagonal make the matrix D, which is p inverse ap is diagonal. Okay, these are the current PM for cps diagonal on me. Sorry, some time. Well, we see that B has at most two linearly independent Eigen vectors. It's you and the. Yeah. Right. And so like you think like man people needs to be fucked up that they thought that was like a hand. Therefore it follows that our matrix B is not similar to a diagonal matrix, which is the same as a definition that B is not diagonal. Izabal War. Yeah. It's like, oh yeah, nobody fell for them. They all knew like this is just a retarded.


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3 answers
Assume that a sample is used to estimate a population proportion p. Find the margin of error M.E that corresponds to a sample of size 285 with 219 successes at a confidence level of 99.8%.ME%(report answer accurate to one decimal place; answer should be reported as a percent; not a decimal though do not type the percent sign)Answer should be obtained without any preliminary rounding: However; the critical value may be rounded to 3 decimal places_
Assume that a sample is used to estimate a population proportion p. Find the margin of error M.E that corresponds to a sample of size 285 with 219 successes at a confidence level of 99.8%. ME % (report answer accurate to one decimal place; answer should be reported as a percent; not a decimal though...

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