(a) Find the characteristic polynomial of 3 6the matrix A2terms of the variable % _p(~) Find all the eigenvalues and corresponding eigenvectors of matrix -3 6 A = ~...

Find the characteristic equation, the eigenvalues, and bases for the eigenspaces of the matrix. $$\left[\begin{array}{lll}1 & -3 & 3 \\3 & -5 & 3 \\6& -6 & 4\end{array}\right]$$

This problem asked us to find the Eigen values and Eigen vectors for the given matrix. We do this by first finding characteristic polynomial, which is equal to the determinant of the matrix. A minus landed times the identity matrix. This will be one minus lander to six and negative three minus lambda and taking the determinant, we get one minus Lambda native three minus lambda minus six times two. This wisdom calculation will sell that to be you Land a squared plus two Lambda Grannis 15 which is equal to Lambda plus five times lambda minus three. In there are two Eigen values will be negative five and three for the negative five Eigen space we can solve for the Eigen vector since a minus limbed I times X where X of the Eigen vectors equals zero. Consult for this by first playing in negative five that will give us the vector one minus five to six. Native three minus 50 sorry. Plus five. Since Meg decides times the Eigen Vector X one, which is equal to 66 and to to times a one b one which we're gonna be the components of the Eigen value X one, and that is equal to the zero vector. So the only vector that will be able to solve this system is going to be Are I in Vector, which is one negative one. So that's our connector for land vehicle Mega Five for the three Eigen space. So land able to three. We do the same. We plug in three, we get one minus 36 two and negative three minus three times these. Ah, I can vector that is associated with land equal to three. This becomes negative. 262 negative six times a to do you to think of the disease, Director, The only director that will be able to solve this system is going to be our Eigen vector X two, which is equal to 31 And so these are our values and their associated Eigen vectors.

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

This question covers topics relating to the Eigen values and I get better of three times three. Matrix we are going to find all I got values and I got a member of this matrix far. So I will be noted this metric by a first we are going to find a man Atlanta I Right, so it is one minute Lambda, 2, 1- Lambda and -2 321 minute linda. And then you find a determinant of the obtained matrix. Right? So determinant of it is the cubic function. Right? So I just do the computation, I will skip that and you are going to get this right and then you have to sell for the cubic function. Right? So for the equation of the cubic function, This one equals 0. Then you are going to get three different solutions number one. So it goes to one, number two equal to one minus two. I and number three is one. Last two are okay. And so those are three Eigen values for these. Even matrix. Next we are going to find the Eigen better. Right, so cover responding to the wagon value λ one equal to one, then I will flock in and half a minus I. Right, so the matrix is that and I just simplify it and I'm going to get this matrix Okay, right, and then I'm going to find a new spate of this matrix. So that means I'm gonna solve for the system of equations I asked equal to zero. And so so that which is called a new space of this matrix. Then I'm going to get the first Eigen factor B one. one. Maybe 1.5 and one. Okay. And by the same method you are going to do for the second Eigen value, you do the same thing and then you are going to get the second Eigen vector. Right? So the Eigen vector is zero, negative I N one. Okay. Right. And lastly you do for the last wagon. Better one was do I? And then you can feed three equals 2 0, 1. Right? So those are the Eigen pairs or the even matrix? We finished.