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Problem 1.(2 points) Find 2 * matrix such that[3]' and [-1]are eigenvectors of A with eigenvalues 9 and 3, respectively:...

Question

Problem 1.(2 points) Find 2 * matrix such that[3]' and [-1]are eigenvectors of A with eigenvalues 9 and 3, respectively:

Problem 1. (2 points) Find 2 * matrix such that [3]' and [-1] are eigenvectors of A with eigenvalues 9 and 3, respectively:



Answers

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrr}2 & -1 & 3 \\3 & 1 & 0 \\2 & -1 & 3\end{array}\right]$$.

This problem asked us to find the alien bodies and Eigen vectors for a given matrix. We do this first by finding the characteristic polynomial, which is found by taking the determinant of the matrix a minus lander times the identity matrix. So that will give us the polynomial negative two minus Lambda Times one minus Landau Times Negative three months Lambda minus one plus deeply turns negative. Two minus lambda minus negative. Three Minus lambda which will solve out to be five plus four lander plus lander squared times Negative. Lambda minus two equals zero. Then solving out for Landau, we find our Ivan values Teoh equal Negative too negative. Two plus I and negative two minus side, then self Eigen vectors We want to solve for vectors X such that a minus land. I times that I correct X is equal to this year. A vector. So first will do this woodland equal negative too. We plug us into a months Land I, which will give us the matrix 011 113 and zero Negative one negative one times Eigen vector X one And this should equal this year a vector. So as we can see since Ah, the top row 010 That means that our component B one has to equal zero. As a result, we see for the second or third row that we have one a negative feeling for the components at a one and C one. Meaning that a 11 and see what a legal one are. Sorry. Yes, well, even one. And that gives us our first Eigen vector to equal 101 Then we have land equal to native to Plus I were you the same thing. We plug this into the matrix that gives a snake that I won one 11 less eyes three and zero negative. One native one minus I times x two equals which well, then equal the zero vector, then solving out for this system of equations, which is going to be a little bit tedious. But by some things out, you should find that X two is equal to two months. I one plus two are and one. Lastly, we do the same for Lambda. Equal to negative two minus side. Putting this into the matrix. We get I born one 11 plus I three zero negative one native one plus I times x three equal to zero. And that gives us our third again vector to equal two. Plus I one minus two. I won and those are final answers.

In this video, we're gonna go for the answer to question of a one from captain 9.5. So it has to find the item buys nine vectors off this matrix A starting with the values we need to find the determinant off the matrix. A minus. Ah, I Where are is Predetermined s So this is gonna be Yeah. The determinant of the matrix minus four minus R 22 minus one minus. Huh? This is gonna be equal. Thio. Ah, the top left ties top times bottom. Right? Uh, the munchies, counsel. So we've got focus. Our times one plus R minus, bottom left top right, which is four that's gonna be squared close five. Uh, then you push for minus four. No factories. This thio, Uh, plus five when you find the bodies of off which thy zero s So let's just stop in the top right here. So therefore, from this, the Eiken values are gonna be I was equal to minus five. So, brother again, minus five. Well, zero. Okay, so the I convict us. Which associates with jack value first for ah, for eyes ableto minus five. We need to look at the vector and so sorry. The matrix A minus I If our ways minus fires that matrix a plus five I times what we're gonna calculate to be their item. A factor you want. Who needs to find Thea compactor? You won't touch the Vatican, Zira, A case of writing out this matrix we get with that one, two, 24 times by You want sequences? Aargh! So therefore you want we could set her equal to? Well, we need the top component to be minus two times the bomb components would make the bomb. Capone You could, you remind us once said the top burn to be too. Okay, let's find a second. Like a value. We did the same thing, but when it's equal to zero, so it's just gonna be a minus zero times. I does that. Didn't see my tricks. Talk to you 200 Hey, was just minus for two two minus one times. You too. You zero. Looking at the bomb components off the matrix. We can choose you two to be able to. Well, we need the bomb component of you two to twice this operation. Futile because my apartment is one. We can let the bomb would be too. And there are

Hello there. Okay so for this exercise we need to find a matrix A. That has the following Eigen values. And again vectors let's remember that when you have different Eigen values and different Eigen vectors for searching matrix you you can diagonal eyes that matrix. Right? So basically if you have your matrix A then you can obtain a diagonal matrix if you multiply a matrix speed times P inverse times A. And times another matrix speed. Okay, so this matrix C was diagonal and this matrix P was some convertible matrix in particular. This matrix speed was constructed Putin Here. The iron vectors. So in this case is one minus 11 110, 1 -10. Okay. And of course from this we can obtain the members. So the members of this matrix, I'm going to put it directly here. So the inverse of this matrix is 001 here. One health Times The Matrix 002, 110 and one minus one minus two. Okay, so there's the embers of B. Now when you do when you multiply this matrix A. In this case we don't know you obtain a diagonal matrix but this I have no matrix. What it have is the correspondent Eigen values for the correspondent Eigen vectors in each position. Okay so in the first column we're going to have The Eigen value associated to the first column. Right so in this case is one 00. Then in the second terms of this diagonal matrix we have the following Eigen value for the correspondent again better in the position to in the column two of the matrix. So here is minus one, 0000 because the other I can value is zero right now you can observe that we can perform here some operations to obtain a Basically we can multiply by P to the to the left. So in that case we obtained here p times a diagonal matrix use equals to a times the matrix speed. But we're interested in a So what what we can what can we do here is multiplied by P the members to the right of the expression. So we we and we p times the diagonal matrix times PM burst is equal to a or similarly there's just a change of the sites in the in the inequality is that a matrix that we want to find equals to multiply the matrix B that contains in the columns the correspondent Eigen vectors, they're gonna matrix contain the Eigen values and the members of this matrix speed. So that means here we have this matrix speak with the Eigen vectors one minus one, one, one, 10, 1 -1, 0 here. Then the diagonal matrix The Diagonal Matrix was equal to 1 -10. And the embers of B was equal to one half. We take a common factor out And then the Matrix 002, 1, 1, 0 on one -1 -2. Okay, so we need to multiply these three matrices and at the end we we have the following matrix for a this one half, we keep it out times the matrix minus one minus one, one minus one minus one. I'm sorry. Here is to minus one minus two And 002.


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