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Let An [ij] be an "matrix and let bn be veciol of size such that[6+4+3]Qn =1 and(a) Use Maple commands t0 generate the matrix A; and the vector b, (b) Solve th...

Question

Let An [ij] be an "matrix and let bn be veciol of size such that[6+4+3]Qn =1 and(a) Use Maple commands t0 generate the matrix A; and the vector b, (b) Solve the system A,= Dn tor Guess the general form of the solution for arbitrary:

Let An [ij] be an " matrix and let bn be veciol of size such that [6+4+3] Qn = 1 and (a) Use Maple commands t0 generate the matrix A; and the vector b, (b) Solve the system A,= Dn tor Guess the general form of the solution for arbitrary:



Answers

Let $A$ be an invertible $n \times n$ matrix, and let $B$ be an $n \times p$ matrix. Show that the equation $A X=B$ has a unique solution $A^{-1} B .$

Is it a question over 45 Foster for finding determinant of the matrix stuff you're dealing will do. Your post entry is already view second, and this would be negative. So your negative one times your negative one my history last two times your negative to multiply by minus one. That is to unless one your day is equal to negative one times negative for and two times three of your four plus six and which is equal Do then. So where do communities then? Now find all minors off the metrics. So here I am 11 which means eliminate postural and first column. So you're a memorable one. That is 91 three negative two and one and which is equal to five. Now. I m one pool, right? We Which means eliminate first row and second row. Sorry. First row and second column. Right. So your first choice zero went toe and the second column ease one minus one minus two. Now, any minute dead of your negative one treat and 11 which is Nick will do 94 and 13 which means eliminated for straw and Third Column. So here magnet E one negative. 11 minus stool, which is equal to three. And to one that is eliminated. Second ruined for school. Um, so you're want to minus 21? We could you call do five Now I am double toe. That is Eliminate. Second row in second column. So amicable toe Teddy zero to 11 which is equal to negative two. Now, I am to three. And these eliminate second row and third columns for your 01 And you're one minus two. The visit will do minus one now n 31 that is eliminated tundra and for school. Um right, So you're one too negative. One trees, which is equal to five now, and three to that is eliminated down. Draw in. Second column. So here entered Recio tu minus 13 We really do. You're the 13 that these any major town ruin. Todd column years, you know, minus one minus one minus one. Majestic. Going to one. Now Write all miners and the main picks. So here. My minus for three by minus two, minus one. And you know, if I too when one that is three your medics off miners. No, Jane, We sign off. This man picks according to these sign medics, right. Which means we have to only 10 to sign off here. Negative for I. You're too in here. Negative one. By joining this line off these four elements that used the perfecter off metrics EMC I, your emcee that he's 543 did you find to get you to one? You're fine. Minus 21 right? No, that is the perfecter of metrics. No, a joint off a issue and by passports off EMC right? Public transport go This metrics that used the journal faith. But he went flying for the treat. Negative five negative to one. And here by minus two one. Right. That is the enjoy coffee. Now, why in the investment weeks So you're a anglers East you, and by one by d my reply right and joined offer right Our new communities. The that is 10 and right. The engine off metrics say which we have already found. You're finding word in Britain that is one by do negative for anybody. But then that is negative one by do I've been fighting with it better. And that is one by two. You're do it by fine. You're one by fight. I've been here one by fly, your three by 10 year one bite end rain here one by 10. This is our interest mentally. Thank you.

This is a question number 40 1st of all, find e determinant of the medics. So here, minus one, Marty, by by one that is minus one and your four multiplied by minus two, that is minus eight. So you're easy. Quarter minus one. Therefore, the communities seven, right, No find. And the bottle on which means eliminate first royal for school. So your end about one is one, similarly and want to his four and 21 is negative. Two on and about Louise. Negative one. Right now. Write all these foreign tears in the Matrix. You're one foreign negative to 91. Right? That gives the medics off minors right now. Jane, the sign off the med makes according toa last minus minus plus. Right So far. Medics off protector, that is N C. Ease. Year one minus 40 or negative? Negative. Was it used here too? The year 1981. But these this is the conflict or metrics and see right, well, joint off a issue. And by transpose off and see right, But he all right, trying off a unit for do here, you know emcees one minus four. So our colonies negative one minus four Now The second Roy's Tu minus one, right through our second Balinese to minus one. Right. But this is our joint off. A no endorsement makes is you. And by one, by determinant, off a multiplied by I joined all fate like So you're the easy called 07 Find the majorities. Want toe? No go do for Mega Do you want so in worse matrix these one place they wouldn't. You're grew by seven your integrity for by seven and your negative one right there. This is our investment weeks. Thank you.

Everyone. So the problem were given. Today starts us off of the matrix A equals to negative to 14 three negative seven to planets and upper triangular matrix. Because everything below the diagonal a zero and first things first. It wants us to find the Jordan canonical form. So we're going to know that the diagonal of the Jordan canonical form is gonna be the Agon values. Let's find the item values. So to do that, we set the determinants of a minus, slammed I to zero and solve that characteristic polynomial for lambda. So a minus lander, I that's going to be tu minus land. Uh, negative to 14. That's a 10 14 0 three minus Landau. They have 700 to minus lambda. Okay, we need to take the determinant of that and set it to zero. Well, what's the determinant of three by three matrix? It's this number times the determinant of this matrix, right? Plus this number tons the determinant of this matrix. But because it has two zeros here negative, sometimes zero minus two money slammed. A times here is gonna be zero. So that counts for nothing. And similarly, this sum is going to go to zero because the determinant of this matrix and blue times 14 the determinant with matrix and blue is just gonna be zero right? Three of its elements are zero, and they're all multiplied at least one other and zero time 00 So the only thing we have to worry about when this determinant is this first green part. All right, so what's that? That's the ah to minus Lambda Times three, minus lambda Times two minus lambda Because it's times this minus this products. But that products gonna be zero. So it's just that it's just down the diagonals that actually works for any upper triangular matrix. So just a quick trick. I did it out just to show you why it works because, uh, I'm not sure if the theorems in your book, but, um, it probably is somewhere, but sometimes it's just helpful toe realize why you have what you do, and it is pretty easy to show. So that's gonna be zero. So we see that Lambda one equals two isn't Eigen value, and that has multiplicity of two because it's twice in this characteristic polynomial, so just write em one equals two and Lambda too cools three. And that just has until vehicles one multiple cities have to add the three right, because it's a three by three matrix. Okay, so the next step is we need to know how many join blocks there are, because we do have one Eigen value with multiple Steve to use that could either have one or two Jordan blocks in it. So we need to find the number of linearly independent Eigen vectors in the Eigen space or window one equals two because that's going to be the dimension of the argon space is gonna be be number of Jordan blocks for the ag value. So the way we find Eigen vectors is we do nt minus two. I times 50 will call V one because you know that Lambda two has an Eigen vector associated with an equal Sterno. Okay, so a minus 22 I, that's going to be 010 on the diagonals. It's your 10000 scroll up here. That's negative to 14 and negative seven, and we need to solve this for 000 right? Just set up in augmented matrix. Now it's easy to see that this row is negative two times this row. So they just cancel out and we actually get the very simple matrix. Um, 01 if seven. All right, quick and simple. And that means that r V one, we solve it. So the second component, minus seven of the third component, needs equal zero, so we can make the second component seven times the third component just like that, then our first component can be anything we want. So we're actually gonna We can multiply these each by any number, right? Say times a and then the the first component can be anything. It doesn't have to be a otherwise would be one times a rights we can write B and that actually, we can split into two vectors. So let's call him. He won a and he won't be. So. You wanna a equals a time 071 and b one b equals be times 100 Right. And if you have these together than you get few one. So this is there to Eigen vectors for this Eigen value of two. So that means it is actually non defective. Um, it has the dimension of two and a multiplicity of to. So that's perfect. So we can just write out our, um, Jordan canonical form. And because it's not effective, it's just going to be a diagonal matrix with the, uh, Eigen values down the middle. I'm perfect. That's it. Um, there you go. We have a joint economical form. But then the question asks you to find the transformation Matrix s such that we can get J from a by doing this right. And so you know how to do this? This is the last section. This is diagonal ization. So because of the Jordan canonical form is just the diagonal form, this s is just going to be a Each column is going to be the Eigen vector for the Eigen value that it lines up with. So we know the 1st 2 Collins, they're gonna be V one and V one b. So we need to find the Eigen vector for Lambda equals three. So, again, that's a minus three I. So that's going to negative 10 and negative one on the diagonals. So native born zero negative one zeros. Just make sure I got it right. Let's scroll back up negative to 14 and snacking seven. Perfect. And this is gonna go to zero, and we need to solve this, and we'll call it V two. So, uh, we need to solve this and find the vector. So that is going to be, well, these air just linear multiples of each other. So they're gonna cancel out, Um, negative one negative. 20001000 And we can erase these negatives just because we don't need him because we can multiply the whole thing by negative one, right? And then that's that's just it. That's in real reduced form. So we can write the two equals. Well, the third element has to be zero, so you can just put that in there. The first element has to be negative two times the second element. So we can just put one and negative, too, and then that can you can multiply that by anything so we can call it. See, it just has to be of that form, all right, because zero times sees just gonna be zero if we plug it into our A minus Lambda High times V two equals zero. So that's our third again Vector. And now we have to put them in the order in which we listed are Eigen values in the Jordan canonical forms. That's 22 and three. So we need to do that to their that their the first the 1 81 b and then B two so as equals, if you want a 1071 and then 100 right, and then V two is negative two 10 Perfect. So that's our s. And that's all we need to know. We don't need to calculate the inverse, although I highly recommend it just to check your work if you're doing it yourself. Obviously, I've given you the answer here, but for future reference, and you can check my answer. If you multiply a out as such, you should get back this this J and that's if you've done that. It's hard to have done anything else around. So a good litmus test at the end. But yeah, that's the problem. I hope you guys learned something. I hope this is helpful

This is a question Number 42 now Masterful. Find the determinant of the Matrix D. It's easy, quarter. We are too much deeper warnings to five multiplied by one. Sorry, five months of my B negative one that he's negative. Five. Here we have negative three. So maybe posit you you're too money by way to its more. And fire multiple of zero is zero. And here throw Gondry's already zero. So you're really easy. Quick to bless. Five Last three money by by four. So the easy quarter of your five plus two is seven and sound. Marty, if I do, that is folding. And the former number three. So the determinant of the Matrix in the 26 right mile fine end about one, which means eliminate first raw and first column. So two by two matrix he's 15 negative 12 on day, determined off this two by two matrix These we are to blast five ladies selling similarly, and one do, which means eliminated first row and second column. So on 12 is 25 zero to on the determinant, please, for I am 13 that is eliminated, plus trying Tom columns in year one. Fetal minus one. So I'm one threes minus two. Them to one, Which means eliminating second row. Impossible. So I'm doing is minus 30 of you. Never do you want to. So I'm to warnings they had six them double do, which means unlimited second or in second colon. Formidable police to Theo. And which is it gonna do for? And two or three. I live in a second. Go on a column withdrawal minus three and do a minus one 10 countries. Negative toe for entry. One between. Eliminate going for schooling. So entry when there takes the news. Negative. 30 one. Here, 15 Chizik point Negatives. This dream. No entry toe. So any minute turned role and second problem. So anti police to zero will buy, which is a global trend. Now end about three weekends to minus three one. And what is it you think like no fight for minor Sinden. It makes so you're my tricks off mine or buddies? Say one four minus two negatives. Thanks for my student, Morgan. You 15 then and eight, right? No. During the sign off, women rates according food, This Mezrich Well, that Duthie Perfecter make its NC. So this is a long conflict on that Drinks EMC right now. A gentle Fay issuing by transpose off course Vector matrix, Right. Stands for joint or complex A knees. You say 16 What effect you, you know, will be far or no good, Jane, we're like a room two teenage right now. Inverse matrix A Morse is doing by one by de German it off a multiplied by Joe. Both a right love your determinant off the metrics is 26 Invite the a gentle a Did you have already found? Think b 76 Negative 15. Your negative for for negative turn your negative due to rent. Right? So our English small thinks he's selling by doing 26 through by smoking getting by 26. We're gonna do my parting through my talking Well, nobody fired by told Jean I wouldn't want by Jim for letting one bite harder and harder for my business are dangerous. Don't thank you


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