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For each of the following; give an example of simple matrix which has given characteristic polynomial charA(1) and minimal polynomial min,(2) Hint: the MSTC: should...

Question

For each of the following; give an example of simple matrix which has given characteristic polynomial charA(1) and minimal polynomial min,(2) Hint: the MSTC: should all be upper triangular. Anld should be obvious what their minimal and characteristic polynomials are Note: You do not need to justify YOur answers TouI answer can just be the matrix(a) [1 pt]char4(1) (r - 1)(r + 292(2+3)min.(2) (2 - 1J6 + 292(2 + 3)(b) [1 pt]char4(1) = (2 +I)(r - 2)( + 3)minA(r) = (2 + I)2 - 2)61 + 3)(c) [1 pt]cha

For each of the following; give an example of simple matrix which has given characteristic polynomial charA(1) and minimal polynomial min,(2) Hint: the MSTC: should all be upper triangular. Anld should be obvious what their minimal and characteristic polynomials are Note: You do not need to justify YOur answers TouI answer can just be the matrix (a) [1 pt] char4(1) (r - 1)(r + 292(2+3) min.(2) (2 - 1J6 + 292(2 + 3) (b) [1 pt] char4(1) = (2 +I)(r - 2)( + 3) minA(r) = (2 + I)2 - 2)61 + 3) (c) [1 pt] chara(r) = (r+)2 +2)"(r - 3)3 minA(r) (2 + I)2 + 2) 312



Answers

Find the characteristic and minimal polynomials of each of the following matrices: (a) $A=\left[\begin{array}{rrr}3 & 1 & -1 \\ 2 & 4 & -2 \\ -1 & -1 & 3\end{array}\right]$ (b) $B=\left[\begin{array}{ccc}3 & 2 & -1 \\ 3 & 8 & -3 \\ 3 & 6 & -1\end{array}\right]$

Yes. Were given a matrix and ratifying the characteristic and minimal polynomial of each matrix. That's the thing about in part A. We are given the five x 5 matrix a. But there's nothing with this one With entries to 5000. Yes side. Yeah. 0-000. That you Yeah 004- zero. Yeah 00350 And 00007 is much better than any of the they've been good notice that A. Is block diagonal with diagonal blocks. Day one. This is the two x 2 Matrix 2502 A two. This is the two, x 2, Matrix 4 235 and A three. This is the one x 1. Matrix seven. Okay. Okay. Actually something you said they told me they want so to find the characteristic polynomial trying to character at this point on you, all of each of these diagonal blocks. While the first two blocks are two x 2 matrices and the characteristic polynomial delta A one. Well, this is t squared minus the trace of a one which is four times T plus. We determined of a one which is four the return of mhm. Which we can actually factor as t minus two squared. Likewise, the characteristic polynomial of a two, again, a two by two matrix is t squared minds. The trace of a two, which is nine times T plus the determinant today to which is 14 cause I can't it's a sex which we can factor As T -2 -2 Times T -7. and finally the characteristic polynomial for a three since it's a single tin Is simply T -7. Yeah. Yeah. Right. And therefore the characteristic polynomial of a Well, this is going to be the products of the characteristic polynomial of the diagonal blocks. This is T -2 cube. Right times t minus seven squared. Yeah. Listen to find the minimal polynomial. I'm gay. Let's find the minimal polynomial of our diagonal blocks. Yeah. Well, clearly the minimum polynomial of Bagna block. A one has to divide characteristic polynomial of a one and also has to contain each distinct factor. So it has to be either T -2. Four Ends of a. one of tea is the same as delta. A. One of T which is t minus two squared. Yes. If you test, mm, said a one of a one or by definition this is a 1 -2 I which is clearly not to zero polynomial. Your stolen five in the F four eggs. Therefore all those that are member polynomial for a one is the same as are characteristic polynomial Which is T -2 squared. It's likewise the middle point. O Neill a two is either Uh T -2 in his breast. This is or G M t equals T -7 or are characteristic polynomial way too T -2 times t minus seven. Yeah. No. Now if his tests F. of a two by definition is a to -2. i. Yeah it's impossible. She's clearly a non zero matrix and gov two mm. A definition is 8: -7 i. Which is also clearly a non zero matrix since we have two and 3 on the off diagonal. But usually therefore it follows that are characteristic polynomial. It's very minimal polynomial. A to F. T. Is the same as the characteristic polynomial Which is T -2 Times T -7. Yes. Finally The only option for the name of polynomial day three Is the same as the characteristic polynomial for H2 regions Which is T -7 and therefore the minimum polynomial of a block diagram. Matrix A. Yes. Well this is the least common multiple of the minimal porno meals of our diagnosed blocks. Mhm. In the b. Oh Mhm. This is the waist down multiple of for you t minus two. Yeah squared closer T -2 times, T -7 & T -7. I always say the middle part. No meal will have to contain the factors T -2. Yes squared and also large. If you T -7 old ass pieces, the chances Notice that this is slightly different from the characteristic polynomial which has an extra T -7 factors. That's not one. My Dick isn't zero. Okay then party were given the matrix B. With entries 4 -1000 12,000 00310 And 00031 stressful. Finally 00003. Yes. Yeah man. Okay. Yeah. On your head. Yeah asked in Baltimore. Thanks. So this is uh our metrics following a similar procedure to aid recognize that B is a block diagonal matrix with diagonal blocks. It's those it says live storytelling by bam Margera B one Which is 4 -112 E two which is 3103. Yeah actually it's not quite correct. We'll have a three x 3 matrix instead. Hello my 310 031 and 003. So been a rough neighborhood now Right away we can tell the characteristic polynomial for b. one. This is T squared minus the trace of B one which is six times T Plus the determined to be one which is nine Or T -3 Squared. Do you have a general addition? Didn't click now on the other hand, Find the characteristic polynomial of the two. Regular. Not bad. Still man. Yeah. Experience. It is 100. I love scam artist notice that this is a matrix of a special form. So right away we can actually see that the characteristic point on you will be too Is going to be T -3. Cute saying blacks like times and therefore characteristic point. O neill of B. This is the product the characteristic polynomial diagonal blocks which is t minus three To the 5th power by going there sir. Turn right now on the other hand. And about the minimal polynomial. You know the minimum polynomial is the let's find the middle point of the meals for our two diagonal blocks metal polynomial of B one So Is either T -3 or T -3 Squared. I'll be something. No, we're here. Someplace of course You plug in B12 t minus three. We clearly see that be 1 -3. I is not zero matrix and therefore it follows that. Mm. Oh, polynomial of the one is the same as characteristic polynomial statute. Just driving. Just smart By a special theorem. We have at the memo polynomial of B two is actually the same as the characteristic polynomial of the two, Which is T -3. Cute. And therefore we have the minimal polynomial of B. Well, we know this is the least common multiple of the number of polynomial of blocks. B one, B 2. Yeah. Mhm. This is the least common multiple. Uh right T -3 sq in T -3. Cute. Which is just T -3. Cute. Oh, uh huh. All right. Finally in part C mind makes were given the matrix santa. So hello, What do you know anything about? 3, 2 0000. Just 14,000 00310. Yeah, 00130 And 00004. Now, following the same procedures as in parts A and B. Me. You can find the diagonal blocks of C. Use these to find their characteristic polynomial and use at the time the characteristic polynomial of C. As the product of Characteristic polynomial of the diagonal blocks. And this is T -2 squared child Times T -4 Squared Times T -5. No. It's a crime to look. Mm hmm. Pictures to find a meme a polynomial. Once again, you find the minimal polynomial of these blocks. Did I had my blocks. And you find the least common multiples of his minimum minimal polynomial of the dag block and therefore the minimal polynomial. Yeah. Is equal to T -2 Chinese. T -4. All right. Yeah. Times T -5. Mhm. Very stranger for 10 years. I cannot. I don't need a straight. I don't know what Yes Card. It's cardboard, but like how is it? There's already point. No point playing this way too much.

Look have you given to matrices? A and B. A. is the three x 3. Matrix 4 -22. I can't find it. 6,34, 3 -2, 3. Okay. Here's a question. And the other matrix is B 3 -2. And then the project. Who? Or negative four. It's fine. And it's true too. And and 2 -3. 5. Uh The truth. He's the love that you or over the characteristic of both nature sees delta. T. Is T -2. Mhm & T -1. Where And right behind the minimal polynomial M. Of T. What you can be true. Change Jeffrey Epstein segment It's A for the minimum point of view of. Mt. Remember the fucking like marshals. They commercials cats film version. No, must divide are characteristic polynomial delta. Also each factor of delta. Sure, awful. Since that's the only thing I is also a factor of our minimal polynomial. M F. T. Just therefore it follows the M of T is either T -2 times T -1 or with two. I mean it really just pissed me off in such say mm T is either efforts to, it's for this it's weird because or G. O T. Which is the same as a characteristic polynomial. T minus two times t minus one squared. There are a few steps. Well, first of all we know dealing with our first function at ivy. Kaylie Hanson fear um um a is a solution to it. They a root of its characteristic equation. So that gov which is the same as delta of a. Don't think it was zero. Mm. Hm senses I have a problem but now being test efficacy. Well we had S. S. A. This is a definition a -2 times I. And A -1. Between turn is equal to. We subtract two from the main Bagnall. We get to negative two 26 -54 3 -2 1. And then the most private. Spiked by the matrix pay. Where you remove one from each diagonal. Now we have three negative two 269443 -2 two. And if you carry out matrix multiplication, you'll find out this is in fact the matrix 000000000 zero matrix. So it's called the age of roots of both. S and juice. Therefore this method, the smaller degree followers that are minimal polynomial mot is FFT which was T -2 times 2 -1. If you want to use in fact multiply get t squared minus three T plus two. Agius magician. Let's have some. Okay. Yes. Yes. Now that's only part of it. About B. Well, once again and they could be Gsc. This is the same as delta P. Between no is zero by create hammocks. Yes, the only the test. FFT. Well we have FFB. Once again this is a B minus two. I can be minus I. And first matrix is b minus two. In the diagonals. With one negative 2 to 4 negative 66 2 -3. 3 times of matrix to negative 2-4 -5 6- 83. Poor rents is not. If you carry out major transportation, you can really tell from the first term that's not zero matrix but you get negative 22 negative two, -4 4 -4 and -2 2 -2. Which is clearly a non zero matrix. Therefore the characteristic polynomial can't be fFT because Major B has to be a root of detective upon of the minimal party on the line. Therefore it follows that N f t is the other function G. Of T. Which is the same as T -2 times 2 -1 square. And this is the minimum polynomial of the

We're giving nature sea and ratifying the purchase of polynomial and minimal polynomial which takes right in part A. Were given the matrix I am. This is the five x 5 matrix for one 000 041 00 00410 and 000. You're yes for what? I did it a girl once a day. No that's wrong. started 00041. and the last girl The 00004 suck. Yes. Uh huh. They're looking at. Mhm. This matrix is a few ways to figure it out. He could use exercise 9:30. This matrix has four on the diagonal ones in the super diagonal which one is non zero and zero is elsewhere from that. We can immediately conclude in the previous exercise the characteristic polynomial or sorry, minimum polynomial M. Of T. Is Or I can value let four so T minus or Eigen value or to the his power To those are the two. Yeah. Is that smoking kills? And Jeffrey Epstein was um assad age. There is however another way. So first we'll find the characteristic polynomial. Notice that N. Is blocked diagonal impact the shoes she married down here with diagonal blocks. Yeah. Not only yes. A. Which this will be 10041004. And that's see back which is uh Yeah 4104th. It may stop as opposed. I'm not in the street work for you. It shouldn't be a one here. Uh This should be zero. So in fact this is not necessarily are minimal polynomial. I was wrong about that. It's a little different. Has about zero before. All right. Yeah. Right. She could get it with these block diagonals. Yes, it follows that. The characteristic polynomial delta T. Of a. About the A. F. T. Right. Well, using the previous exercise is the same As our memo point of view. Okay. and eight which is t minus or dag moment for to miss case. I am using the results of the previous exercise kind of like a baby. The characteristic likewise the characteristic minimal polynomial B are the same. And this is T -4 squared. Therefore it follows that the characteristic polynomial of our whole matrix M. Well, you know this is because this is blocked at all. The product of the characteristic polynomial of a kind of the characters of polynomial of B, which is t minus four. Yeah. All right. She's also determined because it's over trying to you just say like you never had. Now on the other hand, we have that arm m applying on your embassy. Well, this is going to be the least time multiples of our two characteristic polynomial of the diagonal blocks, delta F. T and delta dfc. It's it's the least common multiple. Mm T -4. Huge. Okay. Yeah, blurry. All right. Notice that the exponents care in there, minimal polynomial is as high as the largest bloc eggs that was part of And in part the were given the matrix M. Prime. And this is a four x 4 matrix adults had to find 27 0002 00 0011 and 00 -2 four. As we're sort of given a sense here with the way this nature was drawn. It's pretty clear that the matrix M prime set is block diagonal. Yeah, he turned like this right with diagonal blocks like the Thank you. Music scene. A crime most of the two x 27 Matrix in the upper left to 702. And the prime which is the bottom right? About the matrix 1. 1-. 2. 4 for shoes. Now the characteristic and minimal polynomial of a time. Using the previous exercise. This is the same built a crime and M. S. Of a crime T. Each or both T -2 Squared. And likewise delta P. Prime of T. And mm be crying a tear the same. They're both equal to T -1 signs. Oh I'm sorry. That's not true. He's a little bit better. Don't is to find the characteristic point on you will be more of a two by two matrix and services. T squared minus the trace. 5 10 T. Because the dependent of the trees. Six, fucking Which we can factor is 2 -2 T -2 times team on the street. The old which we see has distinct factors. Yeah, tail tail soft. He was investing last and therefore the minimal polynomial and to be prime of T is the same as delta P. Prime of T, Which is T -2 times 2 -3. Okay. Therefore it follows that the uh huh characteristic polynomial of M. Prime. This is the product of the characteristic polynomial Zaveri bicycle blocks, delta, a prime of T and delta decline of cheese, which is equal to three minus two huge times t minus three. We decided middle. I think so. And the minimum point of the old city. Well, this is the least common multiple of the middle polynomial burger diagonal blocks a prime and be time, which looking at these two minimal polynomial there, This is going to be T -2 squared times t minus three. It's a little bit different. Oh, he has the online position.

We're giving them a polynomial and ratifying matrix A. With this new point. Your meal in part. They were given the polynomial T cubed minus five T squared get in the game plus 16 plus eight. This thing, yeah. Smart. All right. Simply take a to be the companion matrix featured in this exercise. Sometimes I just have feeling. Yeah it will be the companion matrix. Which the last column will be. Yeah. The transpose of -8 is -6. Yes. Five. That's how you know, it stands in other words scared we'll take a to B. All right. three square matrix with ones on the sub diagonal and negatives of the coefficients in the last column and zeros elsewhere. So we get a matrix 010 zero 0 1. The columns and the final column is that's negative eight -6 and positive five. Yeah fuse it enough. At least it's going to it starts around five Once you put down and in part B were given the polynomial T. The fourth minus five cube. Yes, that -2 T. for 70 set your belong plus four. The simplest answer would be to choose the companion matrix for this polynomial position. This is the matrix with one's well it's first of all a four square matrix, ones on the sub diagonal, the negatives of the coefficients in the last column and zeros elsewhere. Their first column, for example, is 0100 2nd column is 0010 3rd column is 0001 And our final column is -4 negative seven, two and five, you know. Mhm regrets. Yes.


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