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Point) Find the eigenvalues of the matrix A12 6-8 ~9 andThe eigenvalues are...

Question

Point) Find the eigenvalues of the matrix A12 6-8 ~9 andThe eigenvalues are

point) Find the eigenvalues of the matrix A 12 6 -8 ~9 and The eigenvalues are



Answers

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrr}10 & -12 & 8 \\0 & 2 & 0 \\-8 & 12 & -6\end{array}\right]$$.

This problem Give Gives us a matrix and asked us to find its Eigen values and Eigen vectors. We do this by first taking the characteristic polynomial, which is found by the determinant. Ah, a minus land I, which equals zero. So that gives us the determine of matrix seven minus lambda a zero negative eight negative nine minus lambda zero and 66 Native one minus Linda Um, and that themselves out to be will be 17. Slander The Times Negative nine months Lambda Times native one minus lander plus 64 times negative one month's lambda which solves out to be Lambda plus one cute meaning. And since the should equal zero, that means our Eigen values equal to negative one with an ultra break Multiplicity three. In order to sell for the Eigen vector, we need to sell for a vector X such that a minor slammed I times that Eigen Vector X equals zero doctor for playing. So we're gonna plug in land equals negative one. Doing that gives us the matrix 880 negative eight negative 80 660 times X which is going to be equal to if you subtract this the first surfing the second row and then divide by two. We get the Matrix for negative for three and then zeros all on the balm to rose times a B C, which is equal 000 And from this we see that for a minus four B plus three C should equal to zero. And there's only going to be to, uh, matrices that will that are literally, literally independent. That will satisfy this. And those major cities are 110 and negative 30 four. Giving us our Eigen vectors for the Ivan Value. Negative one, and that's our final answer.

This problem gives us a matrix asked us to find it. Eigen Values and Eigen vectors. We can do this by finding characteristic polynomial, which is found by taking the determinant of a minus limbed I. So that will give us the determinant of negative two months. Lambda three Negative six four minus lambda Taking the disorder that we get negative two minus Lambda Times Reminder Slammed, uh, minus negative six times three and solving that out. Get lander squared minus two Lambda plus 10. Using the quadratic equation assault for a Lambda, we get land equal to negative negative. So two plus or minus square root of two squared, which are native to square, which is four minus four times A, which is one time see sometimes 10. Divided by two A, which is just too, which gives us two plus or minus square root of negative 36. Divided by two, which is equivalent to one plus or minus three. I so are two Eigen. Values are not really it's one plus Syria in one minus three I to solve for the Eigen vectors we saw for a vector X, such that a minus slammed I Times X equals zero. So taking land be equal. Toe one plus three. My first will plug this in. That will give us the matrix. Negative. Two minus one, minus three. I three legged of 64 minus one minus three I times the Eigen vector X one, which is equal to negative three months. Three I three negative six negative. We are sorry. Should not name three minutes three at their ago and times a one B one equal to 00 So Ah, there's a few different ways to solve for this system. I'm going to start by, uh, playing in B. Warning is equal to one. And then if you saw about the system of equations, you'll find that, uh, will be satisfying a one equal to I minus one. And this is gonna be the only solution to this system. I'm not gonna solve it out, since that is just solving system of equations. And so, as a result, we get the Eigen vector to be I minus 11 now for limbed equal to one minus three. I were going to do the same thing, so we plug in one minus three I and that gives us negative. Two minus one plus three i three negative 64 minus one plus three I times picks to is equal to native three plus three i three negative 63 plus three i times a to be to which is equal to 00 in this case again solving out when a plug in b two equal toe one which will help so about the system equations to give a to equal to negative I minus one. And that gives us our second Eigen vector, which is negative. I months one one and those are our final answers.


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