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46 (b) d F 3a e S 3b f - 3c] 29 2h 2il 2a 10b Zc (c) d Se 9 Sh 4 fa Find the inverse of &/ assuming a # +2. 12 5. Construct a 4 x 4 matrix A such that nullity(A...

Question

46 (b) d F 3a e S 3b f - 3c] 29 2h 2il 2a 10b Zc (c) d Se 9 Sh 4 fa Find the inverse of &/ assuming a # +2. 12 5. Construct a 4 x 4 matrix A such that nullity(A) = 3. 6. Suppose A is a 3 x 3 matrix Determine whether Ais inme 21 (a) Ax = 1 has no solution.(6) The solution space to Ax = 0 has basis {(1,-1,0)}. (c) rank(A) = 3 (d) det(A) = ~4 1 nena

4 6 (b) d F 3a e S 3b f - 3c] 29 2h 2il 2a 10b Zc (c) d Se 9 Sh 4 fa Find the inverse of &/ assuming a # +2. 12 5. Construct a 4 x 4 matrix A such that nullity(A) = 3. 6. Suppose A is a 3 x 3 matrix Determine whether Ais inme 21 (a) Ax = 1 has no solution. (6) The solution space to Ax = 0 has basis {(1,-1,0)}. (c) rank(A) = 3 (d) det(A) = ~4 1 nena



Answers

a. Write each linear system as a matrix equation in the form $A X=B$ b. Solve the system using the inverse that is given for the coefficient matrix. $$\left\{\begin{aligned}x-6 y+3 z &=11 \\2 x-7 y+3 z &=14 \\4 x-12 y+5 z &=25 \end{aligned}\right.$$ The inverse of $\left[\begin{array}{rrr}1 & -6 & 3 \\ 2 & -7 & 3 \\ 4 & -12 & 5\end{array}\right]$ is $\left[\begin{array}{rrr}1 & -6 & 3 \\ 2 & -7 & 3 \\ 4 & -12 & 5\end{array}\right]$

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.

In this problem. We want to start by calculating the inverse matrix of a And so we have a formula to compute this. First, we're going to want to find the determinant of our matrix. So that means that we're going to start from multiplying across this diagonal. I wish so by seven by two to get positive 14. And then we're going to subtract the result of multiplying along our other Dagnall eso minus wealth and we get a determinant of two. Also from here, we're going to go ahead and find the reciprocal of this determinant, which is 1/2 and we're going to perform a scaler multiplication along or along our initial matrix A with a few modifications, I'm so our first agonal seven and two, we're just going to switch those two numbers. So instead of seven in the top left and two in the bottom right, we now have two in the top left and seven in the bottom, right and longer other to Daniel's. We're just going to multiply my negative one. I'm So instead of having four in the top right, we now have negative for and our bottom left will now be negative three. And so from here, in order to find a inverse we're going to do is compute arse killer multiplication. So they get to matrix one negative, too negative, three halves and positive, seven halves. And for the second part of our problem, we want to calculate a times a inverse to validate that it is equal to the identity matrix, which would mean that they are in verses of one another. Um, so we can go and write this rewrite our matrices. Here we have 7342 a times a inverse, which is one negative three halves negative to positive. Seven halves. And we know that in order to compute a matrix multiplication, we can go ahead and start with our first row of a multiplied by our first column of a and verse. I'm so you get seven times one plus a negative. Six soldiers write this as seven minus six. And then along our second column of a inverse, we get negative 14 7 times negative too, plus four times seven house, which is going to us plus 14. And finally, we'll do the same thing for our second row of a. But this time we get three times one in her first column, plus two times negative three house so minus three in our second position there. And finally, for our second column of and worse, we're going to get three times negative, too, which is negative. Six plus two times seven halves. So plus seven. And once the simple five days we see that we get the Matrix 1001 which is the two by two identity matrix. So that tells us that the inverse matrix that we calculated was correct.

Okay, So in this question were asked to find the inverse and verify that BB Inler's eagles be inverse being equals in the identity matrix. So first, we're going to begin by finding the inverse of this three by the remain tricks that we have. So, for three by three matrices, the first step is you wanna put the original Matrix right here on the left well line and then put the identity matrix on the right side. Here we have I three because we're dealing with the three by three matrix. And the goal is to get the left side to look like you're gonna need matrix. And, um, after doing all of those rural operations to get the left side to look like the identity matrix right side will then be your inverse matrix. So, Bush, we're gonna do Road three plus two are one and without does is turn this negative two and 20 So in the third world, we end up getting 054201 The next we're gonna do real two divided by two and that turns that's two into one on DSO In row two, we get 0110 1/2 0 the next to a new road. Three months wide road, too. Yeah, we do this so that this five turns into a zero. And so in row three, we end up getting 00 negative one too. Negative. Five over to one. And then we're gonna do wrote two plus road three. I am. This turns this one into a zero, and we can already see that we're very close to getting the identity matrix on the left side. So in row two, we get to 010 to negative to one, and then we could do road three times. Negative one was turns this negative one into a positive one. So we have 001 negative to five over. Two. Negative one. And then finally, our last step. We have bro. One liners, three room, too minus 21 And so basically, that gets on these, the three and the two right here to become zeros. And so finally, right here we have the identity matrix on the left side, and then the right side. This is our fingers. Okay? And so now we're gonna do the second part of the question s o I would not be in worse being. We're just gonna do some matrix multiplication. So we have negative one times one plus one time, zero plus negative one times negative too, which is equal to one negative. One times three plus one times two was negative, one times negative one which is equal to zero. And we just keep on going with this process for each row. And that will give us the identity matrix strong right here and then for BB inverse weird. But once again, you know, the Matrix multiplication. And so we have one times negative one plus three times two plus two times negative to you, which equals to one one times one plus three times negative. Two plus two times five over to which gives us zero. You know, we keep on going once again. And so from this we can see that not only do be in verse B and B the inverse evil, each other, um they are both equal to the identity matrix. And so we have now just proven the second part of the questions

Come in the system of equations. The matrix equation associated with the coefficients one X 2, Y five C. Two X three wide. Eight C negative one X one, Y two, Z times the coefficient are the variables X, Y and Z equals the constance 233. How to solve this. We can think of this as matrix A, matrix B. And then you're going to multiply on the inverse to both sides. So it cancels out the left hand side. And so you get a inverse times the where a inverse is given to us as two negative one negative 1, 12 negative seven negative two negative 531 times B. And so we multiply a role one by column. That's 4 -3 minus three, Which is -2, 24 -21 is 3 -6 is negative three and negative 10 plus nine is negative one plus three is positive too. So the solution to the system is -2 -3, 2.


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