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Write a 3 x 3 elementary matrix that corresponds to each of the following elementary row operations. Multiply each of the elementary matrices by A ~2 -2 -2 -3 verlf...

Question

Write a 3 x 3 elementary matrix that corresponds to each of the following elementary row operations. Multiply each of the elementary matrices by A ~2 -2 -2 -3 verlfy that the product EA is the matrix obtained from A by the elementary row operation. (0) Add 3 times the second row to the first row: Swap the first and (hird rows_ Multiply the third row by ( - 4)and(a) Find the elementary matrix:E=Find the product EAEA =(b) Find the elementary matrix:E=Find the product EAEA =(c) Find the elementary

Write a 3 x 3 elementary matrix that corresponds to each of the following elementary row operations. Multiply each of the elementary matrices by A ~2 -2 -2 -3 verlfy that the product EA is the matrix obtained from A by the elementary row operation. (0) Add 3 times the second row to the first row: Swap the first and (hird rows_ Multiply the third row by ( - 4) and (a) Find the elementary matrix: E= Find the product EA EA = (b) Find the elementary matrix: E= Find the product EA EA = (c) Find the elementary matrix E= FInd the product EA EA =



Answers

Consider the matrix $$ A=\left[\begin{array}{llll} a & 0 & 0 & 0 \\ b & a & 0 & 0 \\ c & b & a & 0 \\ d & c & b & a \end{array}\right] $$ where the non-zero scalars $a, b, c$ and $d$ are from a field $\mathbb{F}$. Let $E$ be the elementary matrix obtained by adding $-b$ times row 2 of $I_{4}$ to its row 4 . Find the matrix $E A$ by subjecting $A$ to suitable row operation.

Whatever you're going to find the elementary metrics. Our first task, you're still fine. Any american markets. Mhm. For a given metrics that is We got too close to metrics with 3121 is the numbers. Okay, let us as you let the one are you doing or something like it? To the elementary matrix. Yeah, sunday. Yeah, you could. Mhm. You came in this one you don't um into the given metrics is the cost of the identity matrix where i is the identity matrix of a too close to. There is 1001. Do what you can see that play metrics as it goes to the inverse of all these elementary ck's. That is you on the news and you're doing worse. A group get It was Now all we have to do is find these you want to do and you gain worse. So we'll start by taking an identity matters on one side That is from 001. And we have to multiply and do the operations on this subject. We'll get our heart tricks which is given to us so you want to work. So the start was start one x 1 the first we can do this We can do you one minus rodeo what I'm targeting right now that I have to make it as the elementary matters of something of this kind. Something around the sort of this this is my budget to do that. Yeah. Once I do this operation. Are you good? Yeah minus one zero. What? Mm. And this earlier The really useful that is 10. The level of whatever I'm getting on this side. This can be different as my first element hypnotics. I'll take the and what's on the after all the remains are known but on the side we can keep it at this hour even now just to go again. What I'll do I'll just copy the same thing so that the space transfer will not irritate us. One minus 10 and one. Good good. Uh huh. 201 This is what we got on the last place. I'm going through the second operation a little bit AR- Who are one on this. So we'll see what we go after doing this operation on both the sides. I'll do it zero minus 21 and the same. Even previous metrics. And okay. Doing the same thing on the right hand. Seven is good. one Little 01 so that you currently mattered the self and this thing I can name it is my idea metrics now both even into even if you do is ready with me, what I can write is you do do you want into the matrix? Is a cultural identity matrix. Hence like we have seen in the previous page also Is equals two universe. Thank you. You're doing most. No. The next question is how do we find the and also this? How to find a divorce? Both a square matrix of cross too. So let us as human square matters city with the elements A. B. Serie and I lose you that the determinant of the That is equals to 80 minus Bc. This into this is non zero. Hence the piano's exist. So the team was will be For a to cross two matrices with determinant that is A T minus Bc. And the remaining metrics. I can just change it like this. D A -7. With a negative sign under -C. This will be the painless. So I'll do the same operation for E. Even and you do Like we have seen even was equals two 0 -101. So by using the formula that we've written on the previous face even in worse will be equal to 1101. You do was equals two 1 -20 and one. If I do the inverse of this we're using that same form that I have right written down on the previous speech. Okay one. What's he doing now? So now I have my even metrics as well as my data matrix. So hence I have both the metrics even and that is what we were targeting Now the second part this includes the you have to reduce it into Roy clinton. That is already yes. To start this part I start with the same metrics metrics that was equals two 321 and one. One word one. We have to do rule operations until and unless we reduce it to them. Identity matrix form. 1st. I do operation on the first road. You multiply. The first proof is one x 3. Okay, 1, 1 x three. Two and one. No cooperation on the second row. My target is to make this one is zero. So what I'll do is our two minus two and 2 hours. Yeah, I'll get 10 one battery one. But so now the next target is I have my one I have my zero. I have to make this one has one And this should be zero. So to do those operations, what are you? Are you on this? So And are multiplied this with three. So what I did 1, 1 x three We learned to 30. One battery into three will be one. one of the work is over Now. The next target is to make the Sun as one. Sorry I have to make it at zero. So what I'll do is ar minus. Are you wetting? And so good. 1 001. This is our identity matrix. So it has been reduced in the Royal pure in form. Now the second part of the question states it is a long question and the second part asked me to do the value decomposition for a jury composition. I have to do it on the same metrics. The metrics was three on what? Okay, we'll start with this. Let me just define what if any do you first a loser a little. Only the lower triangular matrix. Similarly you is upper triangular matrix subject. The matrix A can be written as product of the lower triangular matrix into the upper triangular matters. And this is what we want to find it and you. So I'll start with first I'll try to make it a particular metrics for a particular matters. I will take our given Matics And to make it up. Triangle Matters. I'll do operation on the 2nd row. Hello roto minus Just to make it at zero. I look to 3rd of Roland. So what I did, I'll get 310 And this will be 1 -1 -2 x three. I got a one battery here. Now this is in my upper triangular lower triangular form is relatively simple to get an that is in the lower triangular form. What is it? Is at least the multi class that be used and making I'm sorry in making our humor tricks with the remaining element as once with dream really element has once I brought people here. Okay, so therefore my l metrics can be The remaining elements will be one. This will be zero because I have not done any operation on this side. And this is where I have to put the multiply that I've used. What was the multi class? I will see that to multiply that I've used this poetry. So this same thing will come to save. This is my alma text. No. What are the judges? My aim it takes 3121 Disabled to my lower triangular matrix 102131. Into a particular metrics. That was That whatever we're from. 310 and one x 3. So this is what yeah. Is equals to help into you. Hence the annual decomposition. Not the third part of the question is to compute the indus the geometric side. The matrix was 31 21 No 1st. We'll take the determinant of this determinant of his How much? Three -2. That is it close to one and a non zero number. So in what can be found out and notice of this exist and I can see and yourself. Mhm exist as Determinant of this non zero. So what I'll do is I'll do both the things in the same metrics. So there 3121. Mhm. This third I like their entity matters 10 proceed to one. Nor do operations on both inside whatever operation I'm going to do. I do it on the identity metal soldiers. Sorry first we'll start with this are one. I'll make it as our one by three. So that this becomes fun After doing this operation. I'll get 1 1 x three. No change in the road. Similar thing I get This will become one x 3. This is zero. This is 01. This sort of no my second operation that I can do this I'll try to make this 20. So do operations like I'll do is equals two. R 2 twice. After doing this and that one x 30 the target value And one way to be here. After the same thing I will do on this side. No change in the role roto will become zero minus stupid. They should do a three. This will become one to learn more pages to this just a minute out where. Okay so after reading this again we'll start We'll reach up to this .11 x three zero number three right now was it the one with the U. S. one x 30 -2 or three and one. So my next stop depression is my target is to make this one with three years This one but 3 is one. So water Lewis I do a patient such as R2 equals two three times or two. So that this one where we will go and become as much So after doing this operation I got 10 and the three The sun was going one Same thing when I do dessert for one. No change They're all dumb with the blind birth three and we'll get this So we are pretty close. Not only targeted to remove this one x 3 and we'll get away and was on the right hand side. So to do that I do operations such as urban. Is it close to our 1 -1 Are to Wait three. What this will do is this will remain one this one by three women become zero. No change in the road to Same thing will do little to no J -23 through one now will become one by three. Well done one and this zero and two minus three minus station I say for the solution is minus one. So now this isn't the introduced I'm sorry, I'm sorry this they were being made a mistake. This will become as well. So now this is in the identity matrix. So this is the calls to our angels. Hence a genius is equals two. What the -1 minus two. And and This is what you get for dangerous. Where he was equals 2. 3. What was a equals two. Sorry I forgot. So you want to one To your boot 1? No same thing I have to do for another metrics that I have already done. I'll just add the pages if you have any doubt. You can ask me later also and you can come and will also this is the remaining of the same thing I have to do with the new metrics be metrics. The numbers are changed but the procedure will remain same. Same thing I've done. Just see the pages I found out first that even then that you do here I have together the three also. No the three or 2 even and to be will become the identity matrix. Now in verse as you already know we are discussed. So for all the three metrics I got the universe. For even you do anything. Then for the given matters have to again go into a role to inform in the reduced ridicule and form. So I had to do these three operations and I'll get the identity matrix. Same thing for the L. A. Victimization. There was the Lord and he was upper triangular metrics. And luckily this already has a zero. So this is right now it's having upper triangular form. So nothing much we have to do on this side as you is already given all have to do is multiply with the I. That will be known as our triangle lower triangular matrix and will get B. Is equal to L. A. For this. So the hair factory ization of decomposition is pretty easy. And for the uh inverse purposes this is the biometrics. I'll check further german determine, it is four into two into minus of zero. So I'll get A. That is non zero, hence and was exist once it was exist. I'll do the same operations. Whatever I'm going to do on this left hand side, I'll do the same on the identity matrix side. So this is my first operation. The second operation. And after these two operations, This is my 3rd operation. I'll get this the identity matrix since whatever is on this side, this will become my envoys metrics. Thank you.

So since two matrices are equal, or if and only if their corresponding entries air equal, we can go ahead and replace the four equations in the system by this single matrix equations. So right, wherever we don't have and actually have a second equation, you have two x one plus x two is equal to zero. Well, I have no x three so that I have zero x threes, right? And then factor out our our exes will be a single column matrix r x one x two x three So when we get here is we just look at the coefficients, right to the coefficients in our first row are one negative two and three, right, So we have our matrix would be one negative two and three Look at the coefficient and then we have 21 and well, I have no zero. I have no x three. So my coefficient there would be zero. And then we have again no X ones or zero coefficient there. 70 negative three and then four and then one zero and one. Okay. And then well, this is going to be times the column matrix. We have X. We have x one x two and x three So we times x one x two x three Is that equal to just a column matrix of what we're equal to is equal to negative three zero one and five, right? And that's basically a so here we have where one in a access equal to be Well, our A is just the matrix A. So write this matrix Here, this is a and then times our column, major, this is Times X, and this is equal to be so There we have a X is equal to be so there we have it all right. Looking at part B well again are linear system Here is again So we have three, um x one So it's all right now we have three x one plus three x two plus three x three is equal to negative three and then we have negative X one minus five x two minus two x three is equal to three and then we have negative four x two plus X three is equal to zero. So again we could just go ahead and take our coefficients and put it into a matrix. And that would be a major. That would be our eight. And then we just our exes. Just our while. We have X ones excludes the next week. So we look at our coefficient matrix. Here is three, three, three, Right, And then we have negative one negative five negative two and then zero Negative. Four and one that those are all of our coefficients. So we have in a second role. We have negative one negative five in negative too. And then we have zero no X ones there. And then we have negative 41 Okay, so there's a metrics. There's are a and then times where x one x two x three and that is equal to RBS. Our column metrics here is negative. Three, three, zero. All right, so there we have it. So again we have here a X is equal to be all right. Take care

Mhm. Here we have given a metrics a physical too one 2 3, 4 over our. And metrics even is equal to 10 minus 11 Matrix E two is equal to 0110 and metrics E three is equal to minus 2001 Now we need to compute the product of mattresses, even E. two And a. three. Then we need to verify that there are elementary column operations, which transforms A. To even A. two, a. 2 And there's A two, A. 3. So we first multiply in the mathematics A with these even elementary mattresses. So it is given us follow. So solution is solution. Okay, lettuce. Let us find the product, find the product even. So let us find product even. So metrics a into metrics even symmetric is 1, 2, 3, 4 and Magics. Everyone is given as 10 -11. Now we multiply it and we get. So the first entry in 1st row is 1 -2 is -1, then second injury is two. Third entries 3 -4 is again -1 and four countries 4. So so therefore A. Even is equal to. So we get the product of matrix A and even is a one is equal to minus one, 2 -1 and four. Now we find the elementary column operation Which transformed a two a. 1. So you find the elementary calling operations which transform A two A. 1. So we need to take a matrix A which is is equal to 1234. So the elementary column operation which is applied is C1 is replaced by C one plus minus one times of C two. So it will be So 1 -2 is -1 than it is to Then 3 -4 is -1 and it is for which is nothing. But our metrics even hence it is verified. Hence it is very hard. It is verified that the product A one is same as the limited column operation, which we have used to transform A to even now moving to the next, we have To find the product A. two. So here, you know we no fine product E E two. So The product A two is given us. Mhm. So matrix A, symmetric A. Into metrics. You too, The mattress is 1, 2, 3, 4. And metrics you too is The first entry in row is zero one and the second entry in the second row is 10 The product is so one in 200 plus two into one is too, so it is too, Then here it is, one then here it is for and the last entry in this through in this product is uh three into one is three plus 400 00 So it will be three. Therefore, E two is equal to 214, 3. Now, we find the elementary column operation was transformed A two A. Two, you're fine. Elementary call them operation be transformed mm to e toe. So take matrix A. Is equal do 1234. So the elementary column operation used is C1 is replaced by C two. Does we get? The first column is 2 4, and the second column is voluntary. So this is nothing but our required obtained the product aid. Hence it is verified and it is very hard Now moving to the next, we need to compute the product A. three. So the metrics, he's now we compute we compute for firing the product mm A three, sorry, So N two E 3, metrics is 12341234 And metrics E three, he's given it minus 20 01 So the output is So it will be -2. The first Entries -2. The second entries too, The 3rd entry is -6, fourth entries for therefore a three is equal to minus two minus 64 Yeah, no, the elementary elementary column operation we transform which transformed mm to E three is given us is given us. So we need to find The elementary collaboration. So metrics is 1, 2, 3, 4. No, we apply a column transformation given us C1 is replaced by -2 times up, see one. So it will be minus two, two minus six and full, which is nothing but which is nothing But our obtained product A three. Hence it is verified The adopting correct a. three. And the column operation used his to find the output is same. Has it is verified? Thus, it is our required solution.

Okay, SO apart. A. Here we consider this linear system. We have two X one minus three x two plus five x three is equal to 79 x one minus x two plus x three is equal to negative one and x one plus five x two plus four x three is equal to zero. So I'm going to find matrices a X and p such that the linear system can be expressed as a single matrix equation. A X is equal to be okay. Eso We know that two major cities are equal if their corresponding entries are equal. So we can replace the three equations in this system by, well, a single matrix equation. We can have, um, so the two x one minus three x two plus five x two is equal to a column matrix on the right. Um, and while we could then factor out the x one plus x two plus x three, right, we have X one times. Um, the first column MATRIX so x one Times 291 The column matrix to 91 plus x two times the column matrix negative three negative 15 and so on, plus the X three times a 514 column matrix. Um, and then where we get our matrix here of to negative three five, right, which is just basically all this just the coefficients in our equation. So is our matrix. We have nine negative. 1115 four. Right. So these are those basically are matrix. And then times while times the column matrix x one x two x three is gonna get us back to our system of equations. And then this is just equal to well, to the B is just what we're equal to you. The column matrix seven negative 10 So equal to seven. Negative. 10 Okay. And then well, we're pretty much in the form, right? What we want in the form. A access equal to be which here we have it. Right. This is our metric equation, right? A, this is matrix A right here. Um, this is X, and this is beef. So we have a X is equal to be Where are a is just our coefficient matrix? Um, actually, just are Well, the x one x two x three and B is well, the coefficients on what on what we call to. Okay, so So, Yeah, that's basically it. Part A and part B. Well, we consider, um, a linear system which is now four equations, right? We can take those four equations and put it into a system in the same exact way where we have our X one times our first column matrix plus x two times our second column, actors and so on. So we end up with Well, the coefficient matrix now is for zero negative. Three one five, 108 Um, again, these are just the coefficients now, so we have to negative five nine negative one and then zero. Because the way the last equation is three x squared with minus X cubed plus seven x to the, um forth so we'd have zero. Um, three negative 17 Okay. And in this matrix is fits on. Here is four by four is then times x one the column matrix x one x two x three x to the fourth or x dot x to the fourth, but x x up four. Okay, and then that's equal to the column Matrix here while 1302 Right. So there we have in the form. Um, a times asked is equal to be so There is our matrix equation.


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