5

11F1 11 11 1 1...

Question

11F1 11 11 1 1

1 1 F 1 1 1 1 1 1 1



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.

In this question. Oh, metrics is given to us. We have to compute the determinant of the given metrics. If the determinant is don't zero, we have to use the formula for inverting two x 2 Matrix. To calculate the inverse of the given metrics first we will find the determinant, determinant equals two, one times -1 -1 one times 1. Won't simply find it bigger, determinant minus two. Oh, we have a determinant of a matrix A Not equal to zero which means inverse of the A metrics exist. Now inverse of a two by two matrix. Canticle pally today's metrics A, b, c B and worse It was 2. 1 divided by 80 minus busy multiplied by b minus B by the sea mm. Now we can diet And was equals two, one divided by determinant automatics multiplied by matrix -1 -1 minus one. But we have already calculated the determinants, so minus one by two times, metrics minus one minus one minus 11. Yeah. Now we will multiply it. So we get the metrics one by two, one by two, one by two -1 x two. Yeah, thank you.

In discussion uh, metrics of order to buy two is given as a cause to the elements of fast through our one and one. And for the second row are one and one. Now, first of all we need to calculate the determinant of this matrix. And then if the government is non zero then we need to calculate the universe of these metrics. Using the formula for inverse of a two by two matrix. So first of all we will calculate the determinant of a matrix A, Which will be equals two. Ah Here the um if the first element is a second, one is being second, taliban is C. And this one is the the Gentlemen of is given by a c minus beauty and development of a will be called to here A B, C and D. Both all are one. So this will become one times one minus one times one. He had a determinant of Metrics. They will be called to 1 -1 which is zero. So the gentleman is zero. So there will be no metrics or sorry universe of these medics. And these medics is a singular medic. So is a singular metrics which do not have any universe

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


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