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1 1 VMMI 7 1 5 1 3 1 ! 1 1 23...

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1 1 VMMI 7 1 5 1 3 1 ! 1 1 23

1 1 VMMI 7 1 5 1 3 1 ! 1 1 2 3



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$\left[\begin{array}{rrr}-1 & 0 & 7 \\ 3 & -5 & 2\end{array}\right]\left[\begin{array}{r}6 \\ -4 \\ 1\end{array}\right]$

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

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.

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 universe of the given metrics by using roll reduction method, the given matrixes equals two, which is a three by three matrix having the elements of Castro as one minus one, three zero in the second row, 013. And in the third row, 11 and one. So first of all, we will consider the metrics mm Having a metrics and identity matrix as follows. 1 -1, three 013, 111. And here is the identity matrix of order three, 100 010 001. Now we will use row reduction method to reduce the left hand side metrics to the identity matrix and uh the right hand side metrics. Whatever we will uh Garda metrics by using the same row operations as we will apply on the left hand side and the metrics we will get at the right hand side will be the universe of these metrics. So, first of all, we will use the row operation for uh and for these metrics as Robin stores Robin minus road to. And now by this operation we will get 1 0, it will be 1 -1 one. This will become -2. Here are 3 -3 will be zero. Yeah, these two rows will be as it is 01 and three. Here is 1 1, 1 here on this identity matrix. The change will be here are one minus Otto. This will become 1 0, has 1, -1 will be -1 and 0 0 will be zero And he has 010. And now he has 00 and one. Now we will use the row operation on the road 3rd as artie stores. I'll C -R. one. So here the road operation will be our three stores are three minus R. One. And we will get here is faster and second role will be unchanged. So here this will be zero And one and 3. And in our three this will be 1 -1 which will be zero. And here one minus minus two. This will become one plus two. That will be three here and 1 0 will be one for this identity metrics. The first two rows will be same and here in the third row this will become zero minus one. This will be minus one, zero minus of minus one. This will be plus one and one minus zero will be one. Now apply operation to the road to as I wrote to his stores, I wrote to our two tries of artery and this will give here is the metrics. This will become faster, will be seen one minus 20. And in second or this will become zero minus tries of zero will be zero and one minus tries of three. This will be one minus nine which is minus eight. He had 3- tries of when this will be 3 -3 which is zero. Aarti will be as same as it is. So this will be 03 and one. And here in the identity matics for struggle be the same. Uh And in second row this will become zero minus tries of minus one. This will become zero minus of minus three which will be plus three and one minus tries of one. This will be one minus three which is minus two. This will be zero minus try someone. This will be minus street. Tomorrow will be the same. It is ah So this will become here now we will apply the operation uh on route Ruben as Ruben stores Arvind -1 x 4th of Our two. So this will store Are one. Will store are one minus one x 4th of Our Do. And hence we will get these metrics as one minus one by fourth of zero will be one minus two minus one. Fourth of minus it. This will become minus two plus to this will be zero and he has zero minus one by fourth of zero will be zero And this is 0 -8 and zero Here in the 3rd row, zero 31. And here in the identity matrix we will get one minus one by fourth of three. This will become one minus three by four which will be one by four and minus one minus of -1 by 4th of Well done by 4th of -2. This will become -1 and plus one x 2, that will be -1 over to only. And here the you know minus ah minus one by zero minus one by fourth of minus street. This will become Plus 3/4 And R two and R three will be will remain the same 3 -2 and -3 Here -1, 1 and one. Now we will use raw operation in row two, row two will store -1/8 of Rodeo. And this will give Here, Robin will be unchanged. This will be 100. And here this will be 0 -8/1 -8 times R -1 over it will give it one and this will be zero. And here in the top 003, one On this identity metrics, this will be 1/4 minus 1/2 and your 3/4. And this will become -3/8. So this will be -3/8 -2/-8 will be -1/4 And -3 times minus 1/8. This will become 3/8. And here in our series will be -1, 1 and one. Now we will apply row operation in the rotary, the rotary stores, row three minus tries of road to. And this will uh you Here these metrics will become 100 and here 010 in rotary. This will become zero minus tries of zero will be 03 minus five. Someone will be three minus 301 minus three times zero will be one. This and here in this uh medics, this will become 1/4 -1/2 3/4 -3/8, 1/4 and 3/8. And in rotary this will become -1 minus tries of minus trio it, this will be gone 1/8. And here uh 1- tries of 1/4. This will become AH 1/4. And here 1- tries off 3/8. This will become -1 over it. So we got these metrics uh transformed to the identity matrix and therefore these metrics on the right hand side must be the universe of the given metrics. A. So we got the universe of metrics A. Which is as follows universities, Medics having elements 1/4 minus 1/2, 3/4 -3/8, 1/4, 3/8, one over it. 1/4. And my next 1/8. So this is the university of the given metrics. Now we can check whether this in versus character, not by multiplying this to the metrics. A And if we get the metrics identity metrics of already then this is the correct inverse of that metrics. So now we will multiply metrics 8 to a universe. And we know the metrics is uh having the elements 1 -1, 3 In the 2nd row, 0, 1 and three In two, 01, one and 1 In the universe. This is 1/4 minus 1/2, 3/4 -3/8, 1/4 to over eight Here, 1/8, 1/4 and my next 1/8. Now, by multiplying first row of this aim attacks to the first column of a universe. We will get eight times saying verses Here. This will be one times one x 4 will be one by four. I'm solving this only this right hand side. The calculations are here, so one times one by four will be won by four only Plus -1 times -3 by it will be Plus T by eight and three times one x 8 will be again plus 3/8. And by adding these, we will get here eight x 8, this will be one. So His first element will be one. And now multiplying fast road to the second column of this matter. We will get at one times minus one by two, which will be minus 1/2 And -1, times one x 4. This will become -1/4 and uh three times 1/4. This will be plus three over four. And here we will get minus 3/4 plus three before which will be zero. So the second element here will be zero. And normally applying first row to the third column of cosmetics, we will get one times three by four, which will be three by four and minus one. Times three by eight will be minus three by eight and three times -1 by it will be uh minus tree by it. And by adding these we will get 08, that will be against zero. So the start element will be zero and multiplying similarly at the other rows to the other columns of these matters. We will get the elements here as uh zero 10 From 001. So this is the identity matrix we got by multiplying here to the universe. So here we verified that the universe is the correct inverse of matrix A. I hope all of you go discussion. Thank you.


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