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Ji 1 nldimo Octenmncu ctical 1 1 1 1 1 tor the' 1 JaluAelld urtertacedannwudtnouan 1 uncictnt...

Question

Ji 1 nldimo Octenmncu ctical 1 1 1 1 1 tor the' 1 JaluAelld urtertacedannwudtnouan 1 uncictnt

Ji 1 nldimo Octenmncu ctical 1 1 1 1 1 tor the' 1 JaluAelld urtertace dannwudtnouan 1 uncictnt



Answers

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

In discussion. We need to find out the universal forgiven metrics A which is three by three matrix having the entries in first. True as 111 in second row. As 102 in third row as one minus one and one. So first of all we will consider the metrics as the medics of a here 111 And 102 And 1 -11. And with the identity medics of uh order three. This will be 100 010 And 001. Now we will use the road reduction method to convert this a metrics to the identity matrix and uh these metrics, this identity metrics to metrics whatever it comes, uh applying the same operation, what we apply to these a metrics. So first of all we will apply the operation for the road to As wrote to Bill Store, wrote to -3. So we'll get here as fast through same as it was one, And here 1 -1 are 2 -1 will produce 1 -1. is zero 0 -1. Here -1 and 2 -1. Ah as one. So Archie will be same as 1 -11. And for this identity metrics this will be one 00 and 40 to 0 minus one will be minus one. Here, one minus zero will be 10 minus zero will be zero. This will be Taro as same. 001. Now we got these metrics. Now we will apply uh the operation in Robin which we can apply Iran stores R n minus R. Two. So this will be are we in stores are one minus Arto and what we will get here as one minus zero will be one and one minus of minus one. Well will become one plus one which is to And 1 -1 will be zero. And this here these two rows are same. Do you know my next 1? one And 1 -1 1 for this Identity Matic this will become one minus of minus one will become one plus one which is to And 0 -1 will become -1. Do you know zero will be one and other to Rosa's saying it well -110001. So we got this metrics now we will apply the operation ah to the road three as our three minus R. Three stores are three minus R one. So we will get rotary stores R. 03 Rohan. And we will get here as 1- zero as the first true as it was 0 -11. And here 1 -1 will become zero -1 -2 will become -3 and 1 0 will be one here. So And for these identity metrics R one and R two has seen two minus 10 minus 110. And here zero minus two this will become minus two and zero minus of minus one will become plus one And 1 -1 0 will be one only. So we got these metrics are here now we will apply operation too. Medics are too as a to minus R. T. So we will get we will get here our two minus R. Two stores are two minus R. Three and we will get this. Merrick says 1 to 0. And here zero minus zero will be zero minus one minus of minus three will become minus one plus three. That is to and one minus one will be zero for third row, this will be zero minus three and one. And for this identity metrics Sorry these metrics we will get our two minus R. Three and first two will be the same as two minus 10 and minus one minus of minus two will be minus one plus two which will be one, 1 -1. This will become zero and 0 -1 will be minus one. Tito has seen -21. Right now Now we will apply the operation in a row are one which will be our urban stores are 1 -22 and we will get this metric says one minus zero will be one to minus two, will be 00 minus zero will be zero and second row as it is 0 to 0 and same. Taro 0 -31. And for this matics we will apply our one Minnesota which is two minus one will be one minus one minus zero will be minus 10 minus of minus one will be plus one And other two rows remaining same 10 -1. Ar minus two 11. Now we will apply the operation to the road to which will be our two stores half of our two. So we will apply this operation to road to and we will get here as in the first row remaining same 100 2nd, row zero by two will be 02 by two will be one and zero by two will be zero again Here 0 -3 and one. And this for these medics this will be ah 1 -1 and one In the 2nd row. This will be one x 2. The eyes road to is divided by two and he has zero by two will be zero and here minus one by two For the row three this will be -2, 1 and one. So now we will apply the operation to row three which will be I wrote three stores. Our three stores are three plus triceps are too and here we will get no one has seen 100 and wrote to also same 010 in row three. We will get zero plus tries of zero will be zero minus three plus try someone will be minus three plus three which is zero and one plus three tries of zero will be one. So we got this and for these matters we will get faster and second row Aseem one minus 11 half, zero and minus half. Now we will apply This operation to row three as -2 Plus tries of half. So this will be -2 plus three by two. So we will get here as -1 x two. And now for this element we will get one plus price of zero. So this will be one only. And here one plus tress of minus one by two. So this will be one minus three by two. So one minus three by two will be again minus one by two. So this will be -1 x two. Now we got these metrics and now we got this a matter is converted to identity metrics. And does this metrics must be inverse of matrix. A. So in verse of medics A should be a university calls to here one minus 11 And one x 2. Zero -1 x two. Here -1 x two. one and -1 x two. So this must be the universe of metrics. A. Now we will verify whether it is correct or not by multiplying it to metrics A. Ah We got the universe of medics. E Now multiplying the metrics 8 2 metrics. Universe. We must get the identity matrix. So here medics should be uh medics is 1, 1, 1 and 102 111102. And in the third road is one minus 11, 1 -11. And we got the medics universes one minus one fine. And here 1/2, 0. My next went over to And here -1 over to one and -1/2. Now we will multiply these medics to universe. First of all we will multiply first road to the americans to the first column of medics universe. So we will get here as uh one times one. So this will be one plus one times half. This will be plus half and one times minus off. So this will be minus off. So this will become one here and now we will multiply first through to the second column of universe. 1st 12, 8 to the second column of the universe. This will become one times minus one. So this will be one times minus one will be minus one. one. Time 0 will be plus zero and one times one will be plus one. So minus one plus zero plus one will be will result to zero. Now multiplying this first row to the third column of universe. So we will get one times one will be plus one. One times minus half will result into minus half and one times again minus half. So this will be minus half. So one minus half minus half will result in 20. Now the multiplying similarly the 2nd row of the metrics to the university. Subsequently by column one, column two and column tree, we will get here as zero And here one and here zero. And multiplying similarly 3rd row to the all, all these three columns of medics universe. We will get zero here and zero and 1 here. So we got the product of a and the universe as an identity matrix, so we got a dot universe as identity matrix of order, etc. So we verified that universe is the correct universe of metrics. E I hope all of you got discussion. Thank you.

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.

This video's gonna go through the answer to question number 11 from chapter 9.3. So ask to use real reduction to find the inverse off the matrix. That 11 one 121 Thio three. So So we conform the combination matrix with the identity and they tried refugees. Okay, so if we subtract to you off the top equation from the bomb equation, then we're gonna get zero one that to you, minus 20 Maybe it's gonna be minus 201 on the inside. And if we should bottle subtract one of the first question from the middle equation, that's gonna be zero That's gonna be one on that's going to zero months. Well, on zero on me, the top equation as it is, Savior zero. Okay, so now we get to be a stick in court because on left inside the bomb equation on the middle or after the bottom row of the majors in the middle of the matrix. All the same, which means that the ah, the row is off the matrix linearly dependence, which by their a born in the book, means that er the identity that's all right with me


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