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Find the characteristic polynomial for the matrix8 B =|The characteristic polynomial is (type an expression using xas the variable no spaces between variables and o...

Question

Find the characteristic polynomial for the matrix8 B =|The characteristic polynomial is (type an expression using xas the variable no spaces between variables and operations. if you do not have access to the power button; type x? as XX,and so on )

Find the characteristic polynomial for the matrix 8 B =| The characteristic polynomial is (type an expression using xas the variable no spaces between variables and operations. if you do not have access to the power button; type x? as XX,and so on )



Answers

Find the characteristic polynomial $\Delta(t)$ of each of the following matrices: (a) $A=\left[\begin{array}{rrrr}2 & 5 & 1 & 1 \\ 1 & 4 & 2 & 2 \\ 0 & 0 & 6 & -5 \\ 0 & 0 & 2 & 3\end{array}\right]$ (b) $B=\left[\begin{array}{cccc}1 & 1 & 2 & 2 \\ 0 & 3 & 3 & 4 \\ 0 & 0 & 5 & 5 \\ 0 & 0 & 0 & 6\end{array}\right]$

This is uh were given matrices and the rest of the characteristic polynomial. With each of these matrices Matrices were given our three x 3 matrices. And we're told to use the formula delta T equals t cubed minus the trace of a times T squared plus a 11 Plus A 2 2. Yeah Plus a 3 3 um minus the determinant of A. Where A I I Yeah. Is the co factor of entry? Little ai In the three x 3 Matrix. Mhm. Yeah. Mhm. So in part they were given the matrix A equals 123304. You made it sound yes 645 No it worked. He d talk. Yeah. So first we'll find the trace of a this is one plus zero plus five which is six. Right? Yes. Then the co factor a 11 This is the determinant of the sub matrix 0445 Which is 0 -16 or -16. Don't need it anymore. The co factor a 22 This is the determinant of the sub matrix. So 1365. Yeah Which is 5 -18 or -13. And finally co factor a 33 This is the determinant of the sub matrix 1230 Which is 0 -6 or -6. Yeah. You know bridge Therefore it follows to be some of these co factors A 11 plus 8 to 2 plus 833 Well this is -16 -13 -6. Which is -35. Most of Yeah And the determinant of our three x 3 Matrix A. You can calculate looking at it. So we have a 48 plus No one goes away. 36. Yes. Mhm minus ah Yeah 16 minus 30 what he did Which is 38. Therefore using all this information we have that. The characteristic polynomial delta of teas is t cubed. Yeah so Ben Shapiro or another minus six. T squared minus 35 T actually uh minus 38 said. Mhm. Legal. Then in part B were given the three x 3. Matrix 1 6 negative too. So -3- zero. Mhm. Yes And 03 -4. Hot definitively stated. We'll apply the same method to find the characters victoria meal for this matrix john so first the trace of B is one plus two minus four which is negative one. Yeah The Co Factor B 11. This is the determinant of the sub matrix 203 -4 Which is -8. So will take this case the co factor B 22 This is the determinant of the sub matrix one, negative 20 negative four Which is -4. And co factor B 33 This is the determinant of the sub matrix 1 6 -32. Oh Which is two plus 18 or 20. And therefore the some of the three co factors be 11 plus B 22 plus B 33 is positive eight. And for a three x 3 matrix you find the determinant of b. This is a negative eight plus zero plus 18. Yeah. Uh finest 0 0 minus 72. And they actually were Which is equal to -62. Like a french french. Speaking on the old nazi plugging us into our formula. We get the characteristic polynomial of T. This is t huge. Hey, you're here. Plus she squared plus 80 plus 62. Yeah. So tonight we're at the Rialto theater and then some some Yeah. Ottawa Ontario cap. Yeah. Where I will be bare knuckle brawl. Prime Minister Justin um The 20 we're doing yeah. Do an oiled Greco roman sol

Priest about it back were given matrices and rest to find the characteristic polynomial delta T. Of each of these major seats. In part A were given the matrix A equals 25 for one. I heard cases. Get this off my wife. We'll use the formula for delta T. For a two by two matrix. So this formula says that delta T. For two by two matrix is T squared minus the trace of the matrix. Ems times T. On the pizza plus the determinant of the matrix M. And again this is for I am a two x 2 matrix. I have you get shot. The trace of A we see is two plus one which is three determined to the A. Is to -20 which is -18. It's and so it follows that the Turkish polynomial delta T is T squared minus three teen minus 18. And he's okay. And this is the police. Yeah. Yeah. Mhm. In part B we're giving them any tricks. B equals 7. -3 5 -2. Yeah. Once again we'll use the formula for two x 2. Matrix trace of B seven -2 which is five. The determinant of beef is negative 14 Plus 15 which is one. Therefore the characteristic polynomial delta of T. Is t squared minus five. T plus one. Really get in on there's a secret prizes. Why? Then in part C were given the Matrix C equals 3 -2. You can hang out with us and say whatever you nine negative three. Well really like Or use the formula again, trace of the Matrix C is 3 -3 which is zero and the determinant of the matrix C is negative nine plus 18 which is nine. And therefore the careful polynomial delta T. Is t squared minus zero times two plus nine which is just T squared plus not.

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.

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.


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