5

000 0 WW 8 1 2 1 { 1 1 1 E H 1 0E...

Question

000 0 WW 8 1 2 1 { 1 1 1 E H 1 0E

000 0 WW 8 1 2 1 { 1 1 1 E H 1 0 E



Answers

$\left[ \begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {1} & {1} \\ {0} & {-1} & {1}\end{array}\right]$

In this problem, we have given the mattresses on be an oxidant remind video. The pair of each metrics is the inverse off each other. Let's consider it Mac tricks be find metrics. Be post refined, a productive metrics A and B begin the metrics which is equals toe identity mattress. Now they find a productive be into a The result of markets would be my 00 010 three do which is not equal ist white into Demetris Since the product of too many places on either side are not I'd into demand tricks. The given to my prisons are not in rows on each other.

Okay, here we have the matrix. That's three by three. And that's given by 011 101110 And we want to find its Eigen values, and I can vectors. So the characteristic equation could be obtained as negative. Lambda cute plus three Lambda plus two is equal to zero. And by solving this you obtain Linda. One common to come on three is equal to negative one in 81 2 Now, for Lambda equals two, we can have the following computation. So we want to solve a minus to I times you is equal to zero and we obtained you is equal to t where tease a free parameter and are that's non zero time stuff Vector 111 Now for Lambda equals negative one we want to solve a plus I times you is equal to zero. So in this case, we obtain the matrix of all ones times you is equal to zero and so you is going to be equal to the following vector So negative s minus t t and s where s and tear free parameters. So let's denote that here s and t our free parameters and are not including the zero value. Then we obtain. This can be written as tee times negative 110 plus s times native one 01 So we have a two dimensional Ivan's face for Ivan value. Lambda equals negative one so that I can vector corresponding to the argon value Lambda equals negative one lives in a two dimensional I'd in space and the even value corresponding to the back of a lambda equals two lives in a one dimensional against peace.

We want to use a calculator to find the inverse of a matrix. I'm going to illustrate this with the decimals matrix calculator. So we're going to hit new matrix, identify this as a three by three matrix and then put in the elements 120 which is optional, type -1, -1 2 -10. Want us to find, then you just type Matrix A. And then the inverse. And you can then identify the inverse as 0.20 point 4.40 negative point to 1.4 negative one negative 1.2. Yeah.

In this video, we're gonna go through the answer to question number five from chapter 9.5 were asked to find the values and I can vectors off the matrix A So first find I'm values. Let's find Determine off the vector. Sorry of the matrix. A minus I This is they made at the determinant off bomb minus R zero. They were zero minus, uh, two 02 minus, huh? So expanded over the top row. We've got one minus our sides by the territory, Right, two by two matrix, Which is Ah, squares minus four. So if we set that equal to Zoomer on and by the bracket on the left is equal zero, which gives us, uh, is equal to one or the back on the right sequence zero, which is Either I was equal to minus two or two. So there I am values for this matrix. Find corresponding Aiken vectors. That's first. Look at the Eiken vector. Corresponding. Talking about you one. So we've got a minus one times didn't see matrix times. What? We're going to find the Eiken vector you want. This is gonna be a zero 00 too restricting Born from the leading Dagenham zero minus one, two zero, two Minds well, times you want is equal to savor. So from this, we can easily see that the second or third components gonna be zero. Because they're the only values of the second component of you one and such that these guys both satisfied on that leaves us with just an arbitrary value for the first component. So we can just set it up. Next, the Eiken vector associated with the hidden value. Yeah, I was equal to minus two. So got a month's arse. That's a plus. Two times the identity matrix. I times you on equal saver. So it's gonna be 30 There are 0 to 2 02 two you won't is equal to zero. Therefore you want could be equal to Well, the first component is easy because it's just three because this first row with the Matrix tells us that three times with the component of you is equal to zero. Therefore, that's important. Speaking today right on both of these to tell us that the second circle bonus just the next reach of that. So we set the meet. I want to be people to one, then the Bob. So that's our second dragon vector. Our third I come back to disassociated. The value are people too, so that Ah a minus Thio I is equal to zero. It was that should be a yuan in there. This implies that minus one zero They were zero minus one too. Zero two minus one. What time do you want the Xaver? Excuse me? This middle one should be minus two and they should be minus two as well. Because the Eiken vector. That's because the Eiken value is to therefore you want Thea Aiken. Vector is gonna pay. What? The first component CT. So determined It's just gonna be zero. That servant from the equation from this, uh, from this world matrix on these rows of the Matrix tell us that the second and third components off the item vector the same as each other. So we set the middle. Wanted one, but the bottom or phone and that's on file.


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