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Curtrcm Llte qJux 1 8 vclocin 1 1 1 1...

Question

Curtrcm Llte qJux 1 8 vclocin 1 1 1 1

curtrcm Llte qJux 1 8 vclocin 1 1 1 1



Answers

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

We're given this magic A We're universe first. Me from the convertible meeting room. Without it, the determinant they You could have zero in a but not in veritable works. Check it. A convertible. What do you say? Actually, I read it down here. Let's find a determining a check of the convertible or not. 111 First, I'm gonna high road to buy world one by negative one and had it wrote to So I guess one might one minus one zero. Making one plus 21 No one here. Next I'm gonna multiply growth three by negative one times wrote to I get 111 negative. Q one is negative one. You know, the determining the mortification old the numbers in the pivot in the diagonal interment is clearly not so. Therefore, we can find a neighbor nullifying chambers through the over inside the right chambers on this side. Very eight in the side. You're a here you have the identity matrix for three by three One here is you here alone? Now we're gonna really do until this side here. It looks like this. Once we do that, we will get a members on this side Look for reduced. Well, we already thought before we're finding the determinant. Do it again. First they can about this here. 11 now weaken Can't hold this position here. Negative one. They won negative times. Negative. 101 Negative. 101 Here one. Now weaken. We can scale the throw here. We can divide the group by negative one. We get negative here. Also here. Positive here. Now I can scale road three by minus a few. Added row to cancel this The negative too. Times road Here. That positive you minus one. That's one. Make a few plus one minus one. Two zeros too. You get one Next. We just need get rid of this. We can. He gave the period. Rowing added to the first would be a bit of this one. Here. You never get here from zero and in one one to negative one plus 01 and 101 Now you get a second road out of the first road in negative. One plus two. Just one. You have one plus minus +10 You have minus to plus one. You hear? This is the identity matrix implies on this side. He had a members say in verse, should be one bureau minus one one, minus 12 and minus 11 minus 11

In discussion. We need to find out the universal forgiven metrics A which is a three by three automatics having the first row elements 111 in second row, one minus 10 in third row 12 and three. Ah So first of all we will consider the metrics Here a metrics which is 111, 1 -10 one 23. And here Identity Medics of Order 3, 100 010 001. And so first we will now use the row reduction method to convert these medics to identity. And this one will be converted to another metrics by applying same operations. Then this will be the universe of these metrics. So first of all we will apply the operations on, I wrote to Andrew three as the road to will store wrote to Minister Through 2- through one And the road three will store Row 3- Roman. So here we will get the robot will be as same as it is and road to will be zero to minus Robin. So one minus 10 minus one minus one minus two. 0 -1 will be here minus one. For oh 31 minus one will be zero to minus one will be 13 minus one will be too. And for this identity metrics we will have 100 here and zero minus one will be here minus one as a zero to minus seven. And now here's one minus zero will be 10 minus zero will be zero. Now for oh 303 minus Robin will be zero minus one is minus one ba zero minus 00, 1 0 has one. See So we got the medics after applying these operations now we will apply the operation on, I wrote one as Robin minus row three. That is our 1 -R3. So we will get here as roman stores Are 1- Artery And we will get here as 1 0 will be 1, -1 will be zero and 1 -2 will become -1. And here's second row as it is zero minus two minus one, 012. No for for this metrics we will get here as when one minus -1. This will become 1-plus 1 which is to hear and two 0 0 will become 0, -1 will be -1. And here this second Rintaro has seen -1, 1, 0 And -101. Now we will apply the operation on row three as rotary stores. Ah Twice of rotary plus the road to has twice of row three plus throw to. We will get here as First row same 1, -1. And here a second role also same 0 -2 and -1 for rotary, we will get twice of rotary will be twice of zero plus against zero. So this will become zero Now, twice off rotary will tour price of one plus minus of two. Twice someone will be to -2 will become zero. And here twice off to will be four and a plus plus or minus one will be four minus one, that is three. And for this metrics the faster and second drug will be same. There is 20-1 And -1. 10 here minus one. And minus When ties of -1 will be -2 plus Of -1 will become -2 -1. That is -3. Now twice of zero plus one will be one only and twice of one plus zero. This will become too Yeah. Now applying the operation in rocketry as Rotary will store one by 3rd off or a tree. This will Result into here has 1, -1 And 0 -2 -1. And here this will become 00 and one x 33 will be one. And for this medics this will become 2, -1 -1, 1, 0. And my when my third off -3 will be here minus one and here one x 3 And he had to buy three. So we got this metric says And now we will apply the operation to row one and row two. As the Robin will store Robin plus row three and wrote to will store Rodeo Plus Row three. Now this will result into the metrics one plus zero will be 10 plus zero will be zero and minus one plus one will be zero. And here zero plus zero will be zero minus two plus zero will be minus two and minus one plus one will be against zero. Here are 001. And for these metrics this will become R two Plus -1. There will be 2 -1 which is one now zero Plus one x 3. This will become one x 3 And my husband plus two x 3. This will become a uh minus three plus 2/3. That will be minus 1/2. And here for a road to this will become uh minus one plus minus one. Again minus one minus one will be minus two when Plus one x 3. This will become four x 3 and zero plus two by three will become two by three. Now for rotary wing same minus 11 by three And two x 3. We got this. Magic says this. Now We will divide the road two x -2. So Rhoda will store minus half of rodeo. So this will become 100 0 -2 divided by -2 will become here as one and here zero 001. This became the identity medics. And here this will become one, 1/3 -1/2. Yeah this will become -2/-2 will be one, four x 3 divided by our multiplied by -1 but it will be -2 x three. And here to buy three multiplied by minus one by two will be -1 x three. Here being the rotary a same -1, 1 x three And two x 3. So we got this metrics as identity and hence this matters must be the university of metrics. A. So a universe will be called to Mavericks one, 1/3 -1/3. Here one -2/3. Yeah -1/3. And here -1, one over T. And to over three. So we got the metrics mhm. Which is the universe of metrics. E no we will we will check whether these metrics is character or not correct universe or not By multiplying these two metrics. A if we get the identity matrix then it is correct inverse of medics. So now we will multiply medics a to a universe as the metrics. He was 1111 minus 10 and 123. So one 11. Run My next 10. And here is 123. Multiplying this to the universe. What we got in the last step. This was one, 1/3 -1/3. And here 1 -2/3 -1/3 minus one. Von over three and 2 or three. Now we will multiply these mattresses. So by multiplying the first row to the first column of these metrics we will get here as eight times a universe will be called to one times one will be here one plus Here one times 1 again, one and one times minus one will be minus one. So this will become one plus one minus one. This will become one. Only normal deploying first road to the second column of these metrics. We will get one times one by three years one by three and uh one times minus two by three will be minus two by three and one times one x 3. This will become plus one x 3. So this will result into two x 3 -2 x three, which will be zero. Now by deploying the first row to the third column of this metric. So this will become one times minus one by three as minus one by three. One times minus one by three will be again minus one by 31 times two by three will be plus two by three. This will again result to minus two by three plus two or three. This will be zero. Now multiplying these same way as we multiply the first. True to the three columns of the universe of metrics. A. We will multiply second road to these three columns and we will get three elements of this second row, which will will be 01 and zero. Similarly for this hard road, we will get zero, zero and 1. We can check. So we got the product of metrics A and uh universe which is an identity matrix of order three, So eight times a university's identity medics of Order tea.

In this caution, we have to use row reduction to find the universes of the given mattresses if they exist. And also check the answer by multiplication. So that is considered The Metrics 1, 1, 1, 101, one minus 11. And on the right side identity matrix of order three, 100 010 001. Now we will roll reduce people metrics. We will apply the operation. Are those those two R -R one artist, those two? Our 3 -60 and Mhm. After applying these operations began the metrics one 11 0 -10 1 -20. and on the right side 100 minus one, 10 -101. Now again, we will apply the operation. Aren't those two are even closer to Are those those two are three minus or two? and our three stores too, Our 3-plus 2 times are too on applying these operations we get the metrics 101, mhm 010 000. And on the right side 010 011 minus 12. Now, since we observe that left hand side of the metrics is not an identity matrix, inverse automatic. Does not exist. Therefore, is singular semantics is up singular metrics. Thank you

Okay for this one. We have a is equal to 12 negative. One 011 and zero. Negative 11 So the characteristic equation for this problem is given by negative Lambda Cubed plus three Lambda squared minus four. Lambda plus two is equal to zero. And when we solve this equation, it's a cubic equation. So you will get the three argon values as 11 plus I and one minus. I noticed that these talking visor complex congregants. So now if we have the Lambda equals one, then upon solving the system, eh? Minus I times u equals zero. We will get that use equal t times 100 and for Lambda equals one. Plus I, using a similar process, we end up getting that you sequel to a T times I'm minus two negative I and one no, for Lambda Contra Kit equals one minus I, which is given here. Then that implies that you is equal to it. Turns out that the Egan vectors are also conflicts congregates of each other. So you was simply gonna be equal to a T times negative. I'm minus two guy and one


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