5

1 =1 L0*1L 1 1 sin (a] 1 11...

Question

1 =1 L0*1L 1 1 sin (a] 1 11

1 = 1 L 0* 1 L 1 1 sin (a] 1 1 1



Answers

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

In discussion. We have to find out the the universal forgiven metrics. The given matrixes of uh three by three order which is equals two in the first row. The entries are 11 and one in the second row. The entries are 01 and one in the third row. The entries are 00 and one. Now, first of all we will consider the given metrics as This one metrics here 1, 11 01, one and 001. And here as the identity matrix of third order. So 100 010 001. Now we will use the roll reduction method to convert this. This given metrics into identity metrics and these metrics into whatever it becomes by applying the same operation to both the emergencies. So first of all yeah we will apply the cooperation for the first True as urban stores Are 1 -12 and Artist does are 2 -23. We will apply these operations mm And in subsequent steps So our two stores Our 2 -13. Now what we will get when we will apply the faster operation to these mattresses and we will get here. As first we will apply for the operation for Robin. We will get one minus zero as one and one minus one is zero. Again one minus one will be zero. And uh 2nd and 3rd row will be just same as it is sorry. In second row we will apply our two minus a tree. So this will be 0 0 and 1 0 again one and 1 -1 is zero And 3rd row will be as it is 00 and one. Now for this identity. Matics applying the same operations for Robin in Robin we will apply Robin minus throw too. So this will be one minus zero here. 10 minus one will be minus one And 0 0 will be zero. And for our two we will play our 2 -3. We will get 0 0 1 0 has one and 0 -1 will be here minus one. And tomorrow as it is. So 001. Now here this is identity metrics at the place of metrics A and this one a matrix must be the universe of metrics A. So we can write the universe of metrics A as a universe he calls to. This will be 1 -10 zero, one minus one And 001. So this will be the universe of these metrics. Now we will check that whether this universe's character not by multiplying this to the matrix. A. And if we get the identity matrix uh by the product of these two mattresses, then this is correct. So we will find out the product of A and a universe which should be an identity medics. So this will be here we calculate we are calculating eight times in verse and we will get it as 111, 011, 001. And the universe of metrics is 1 -100 and 01 -1001. Now on multiplying these matters is we will get eight times a universe as here when multiplying fast road to the first column of this universe, we will get one times one, so one times one plus one times zero And one time 0 again. So this will be one time zero. This will become one plus zero plus zero. This will be one only here. And now multiplying the first robot to the second column of this universe. We will get one times -1 Plus one times 1 and one times zero plus one times zero. We will get here minus one plus one which is zero and plus zero again zero. So this is zero. A normal deploying the first row to the 3rd column of the universe. We will get one time zero and one times minus one which will be minus one and one times one will be plus one. Now again, this zero minus one plus one will be zero. So zero for the third element. Now by multiplying the second row of eight to the first column of universe, we will get zero times one, zero times one and one times zero plus one times zero. And again, one time 0. So These all are zero. So this will again be zero and now I'm multiplying 2nd road to the second column of This universe. So we will get zero times -1. This will be zero and 1 times one which will be plus one and one time zero will be zero. So here's zero plus one plus zero will be one. Now multiplying second road to the third column of this universe and we will get zero times mm Here it will be zero times zero which is zero and one times minus one will be minus one and one times one will be plus one which results in zero. So this is zero. Now we will multiply row three to the first column of this universe. You will get zero times one. So zero times one will be zero plus zero times zero will be zero Plus one times 0 will be against zero. So we get so this will be zero. And multiplying row 3 to the second column, We will get zero times -1. zero times -1 plus zero times one. Again one sorry zero Plus one times 1 sorry one time zero. So one time zero will be zero. These all are zero. So here this will be zero. And now multiplying rotary to the column. Third of university will get zero times zero and zero plus zero times minus one. This was zero times zero plus zero times minus one. This will be zero and 1 times one will be one. So this will result into one. So here it is one. So we got the product of a. and E in versus the identity medics of three x 3. And so we can verify that this universe of matter is correct. I hope all of you got discussion. Thank you.

So this question tells us Matrix A is this minus one minus one one and one. And it wants the show that a squared is equal to zero. It was zero matrix. So, really, all you have to do is take minus one minus 11 and one, and multiply it by itself, minus one minus one, one and one. Figure out what you're gonna get. So in the first case, your take minus one minus one and multiply it by minus one and plus one. So that's gonna give you bus one minus one u zero. Then you're gonna take that and multiply it by this column. So again, minus one times minus one is one minus one times one is finest ones. You get zero if you do the same thing with the second row. Okay, First column you get minus one plus one, not zero. And then finally, you do it with one and one in minus one, minus one, and you also get zero. So that shows it. As long as you understand major small vacation, which I'm assuming you do. Um, then you can see that it gives you this Euro matrix

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.

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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