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Adjoin on the right of A, then use row operations to find the inverse Aof the given matrix A2 1 20 3Find the inverse. Select the correct choice below and_ if necess...

Question

Adjoin on the right of A, then use row operations to find the inverse Aof the given matrix A2 1 20 3Find the inverse. Select the correct choice below and_ if necessary, fill in the answer box to complete your choice_O A. A -1 ="The matrix i5 not invertible_

Adjoin on the right of A, then use row operations to find the inverse A of the given matrix A 2 1 2 0 3 Find the inverse. Select the correct choice below and_ if necessary, fill in the answer box to complete your choice_ O A. A -1 =" The matrix i5 not invertible_



Answers

decide whether the matrix is invertible, and if so, use the adjoint method to find its inverse. $$A=\left[\begin{array}{rrr} 2 & -3 & 5 \\ 0 & 1 & -3 \\ 0 & 0 & 2 \end{array}\right]$$

We're being asked to find the inverse of this particular matrix. It's a two by two matrix. Well, remember, our general form for two by two matrix would be if these were all right items A B, C D. So the finding in verse, what we're gonna do is we're gonna do one over the determinant. I'm gonna call that D, and then we're gonna multiply that by D negative B negative, See an egg. So let's apply that to our example. Well, first off, that means we have to find the determinant which remembers a D minus B c. So, in this particular case, a times D would be two times negative one minus b times c. So we have zero times three. Well, this would just be zero and two times negative. One is negative. Two. So to find our inverse, we're gonna have one divide by negative too. Times two matrix, We're gonna switch R and D values. So we have negative one and then two. And where this is the opposite of our signs of being see? Well, the opposite of zero is still zero in the office of the three is negative. Three. So what we now have to dio is multiply each of our terms by negative 1/2. So when we do this, we're gonna have positive 1/2 and then zero in our second row will have positive three over to a negative one. So now we found the inverse of our original too.

Hi. Okay, so here we've got a kind of uh large exercise because we need to find the members of these four by four matrix. So we're going to use the definition of that joint that say that the inverse of a matrix musicals to get joint matrix of a divided by determines of a. So clearly here the easiest quantity to calculate the determinants. So let's start with that. So the determinant of a in this case is equal to one to calculate it, let's do the process. It's not that hard if you cannot use some patterns. So We got this matrix 311 to fight to to 1389 1 3- two. So we can multiply, We can subtract them with the application of some role to another one and the determinant remaining variant. That means that the determinant, if we take that the second row is equal to two times the first, roll minus the second row. This is equal to 13.1 and the second row will become 0100 13 89 1, 3- two. So here we can apply the here we can calculate determined by using the miners so we can choose the second row and we obtain that this is equal to one times. Sorry minus one. The position. So here is -1. Yes, fine as a wife. So here this we have minus one and this minus one is in the position 11 with 22 So we have minus 1 to 2 plus two which is Even so this -1 is eliminated. So we have just -1 times the minor that corresponds to the matrix 111189 And 1- two. And this will become the determinant of the determinant becomes right. We're going to choose the third column for the for these, for the competition of this determinant of this matrix and that means we have one times two minus eight so I'm going to read it. So we have two minus eight Plus two times 8 -1. On minus nine -2 was one. And from this we obtain that this -1 minus minus one Which is equal to one. That means that the determinant of a Is supposed to one. So we have our first quantity determinant of a is equal to one. Now comes the complicated part that corresponds to the joint. So let's remember that the definition of the joint is that corresponds to the cool factor matrix transpose. So here I'm going to look right A So here we have our matrix A and then we can start to calculate the cost factor matrix first. So I'm going to define the k factor matrix by the co efficiency. iJ Okay, so let's start with C11. C j corresponds To see IJN three so see what want means to take in this value and then the determinant of the matrix that corresponds to the minor In this case the position 1 1 which is art. So we we yeah Conserve the sign and we have here. D. 522389 And 3- two. The determinant is it goes to -4. Okay, so for the rest I'm just going to put the this would matrix because otherwise we're going to take too much time to calculate each of these confusion. So 7 1 corresponds to -4. is the determinant of the matrix with a minus sign here -111 because it's a position one to the sum of these two indices is Both. So you must adhere -1 189 1 to 2. And this result into C 13 is the matrix 252 139 132 That result in -7. C 14 Is 252 138 132 with a minus sign here because some of these two Is off. So we have here six, That's for the first row of the key factor matrix. Then we have seen 2 1 the some of this is is off. So we have a minus sign and then here we have the matrix one 3311 39 3 to 2. At this result in three then see to to is positive. And here we have the matrix 111 189 And 1- two. And the determinant is equal. Two miners all. Yeah, I'm sorry. Here I got a mistake. Here is 2, 2, 2. Yeah, here we go. Yeah, that's it then. Here we got C 23 So this sum is oh, so we put a minus sign here. And then here we got the matrix 131 139 And 132. This result in zero. This determinant. Yeah, on C three, C 24 is uh the so here we don't put a minus sign. So here we, we have the matrix 131 252 Sorry, here is 138 On 132. That result In zero as well. Let's continue. C31 is the matrix here, this is even some of these these two industries, so we have a positive the determinant here, we don't change the design of the determinant. And here we have 311 5 to 2, uh 3 to 2. There's result in zero C three to corresponds to minus the determinants of the mate of the same metrics. 111 222 And 1-2. And you can observe that we have to similar roads. That means that this determines equals to zero C 33 We can change the sign here, which is put the determinant And that means the matrix 131 252 And the 132. The determinant is equals 2 -1 & C34, which is a sign minus determinant here. 131 252 And 132. The determinant is equal to one and the last row of work of factory matrix, easy for one this up. So we put a minus sign here And here we have the matrix 311 5 to 2. On 389 the determinant is equal two minus one Then. See for two positive we have just the The determinant of 111 222 um 389. So this is a cost to zero C43 is negative because some of the industry's is off. And here we have the determinant of 131 252 And 139. This is equal to eight And the last coefficient of work of factor matrix is determinant of the matrix 131 252 And 138. The determinant Is -7. So you can see that this is the hardest part to calculate the car factories values. So you put all these coefficients together and you obtain the cool factor matrix of a is equal to the metrics -4-7, 3 -100 00 minus one one And uh -108, 7. Make minus seven. So this is a factor matrix and we have that the joint of A is the key factor transpose. So we just need to transpose this matrix and we obtain the matrix minus four, 30 -1, two minus one 00 70 -1, eight and 601 -7. So this corresponds to the agenda matrix. And if you remember the determinant of this matrix was equals to one. That means that the inverse is defined just by the joint of the matrix which is equals to these matrix here to these matrix. So these corresponds to the members of our matrix A. Mhm.

So to find the inverse of this matrix, the first step is to create a matrix of minors meaning that for each element off the Matrix, we ignore the values on the crime row and column, and then we calculate the determinants off the remaining. For example, assuming we have a matrix that looks like this, Yes, and I say we're working on a so we just gonna ignore the row and the column and calculate the determining for the remaining value. So in this case, it will be e I minus ethic, and you do it for the rest off the values and back to up back to our problem. In our case, we would get 320144 I'm done and then 3211 and a 3310 And then 231 and then 43 for to show two. Very 32 43 32 And then if we calculate each of those who would end up gain three one three, 212 they like 516 and the next up will be the matrix up co factors and all we have to do. We just apply the checkerboard off minuses to the matrix of minors. And it's gonna look like this. Does this arm matrix off miners? And then we just multiply that by the checkerboard. I mean, And then we would get three negative one connective three the neck of to one to connective five, a positive one. Pasta one. They're six, and we're done with Stop. And the next step is called it advocate. And we're gonna try and suppose all the elements from the previous Matrix by swapping the positions over the I know And what I mean by that Here it's some matrix of co factors, then was swapping diagonally like this and this, but notice that the Diana would not change. So 316 was stayed the same. So let's do that. Then we swap connected one into like one of them to me. Then we get needs one down here and next to then we swap 500 of three. We're gonna have three down here and we got five up there. And then we also swept the last last paper just wanted to so too goes down on one's go back, goes up and last but not least we're gonna multiply that by the determinant and to find a determinant. He's the former before it, which is a times E I. What is the FH? Okay, The I minus f g plus I see the age, and in our case, it's going to be equal to four times three money Swstairl minus two times three minus two plus three times serum minus three. Which is that you go to top minus still monastery nine. The secret to one. So we multiply one. But on the bomb makers here, then I would get the same. So this just on final answer.

I want to find the inverse matrix of the three by three matrix zero negative to to 3131 negative to three. Gonna do this by extending it to a three by six matrix, including the three by three identity matrix 100010001 We're going to do grow operations until we get the left side to be the identity matrix. And then the result would be the inverse matrix on the right hand side. So, um, I'm gonna show you one example of the way this could be done. But the operations that I'm choosing to do doesn't have to be in the same order or done exactly the same way as other ways that it could be done. So first, um, let's interchange the first and second rose into the second row first row. So we're just going to change our first and second row around. So now our first row is 313010 in our second row is 0 to 2100 Ther growth remains unchanged at this point. Then we can, um, scale the first row by 1/3. Then we have 1 1/3 1 0 1/3 0 The other two rows remain unchanged at this point. Now, what if we, um, took our replaced our third row with, um subtracting the first row from it. And row three would become zero negative too minus 1/3. That would be negative. 7/3 and three minus one is to zero minus 000 minus 1/3 is negative. 1/3 and one minus zero is one, the other rose one and two are the same. And then what? If we multiply the second row by negative one? Then we have zero positive. One negative. One negative. 1/2 00 rose one in three. I will keep the same for now. So what if we add 7/3 of road to to the third row to replace the third row? With that, it would be 7/3 times zero plus zero. That's zero seven. Third time's one of step in thirds plus negative. 7/3 0 7/3 times Negative oneness. Negative. 7/3. Uh, plus two. It's negative. 1/3 7/3 times negative. 1/2 plus zero is as a fraction negative. +76 7/3. I'm zero is zero, but negative. 1/3 is negative. 1/3 in 7/3 time. Zero plus one is one and rose one. And to are unchanged at this point. No. Uh, what if we multiply the third row by negative three. So three times the third row, That will change the third row. 200 Negative. One negative. 76 times three use negative. Seven hands. Negative. Three times. 2nd 1/3 is negative. One and three times. Oneness three. Okay, keep going Here. We're getting there now. What if we ad the third wrote to the second row with that in the second round? So that would be zero plus 00 plus one negative one plus negative one. You know, I'm going to make us light adjustment here as I see something different. So instead of multiplying road three by three and step above, let's multiply it by negative three. That's gonna change the signs of all these. And now when we add rows three and two together would get 01 zero. Add seven halves with negative 1/2 and we get three. One plus zero is one and negative. Three plus zero is negative. three. Next. Let's, um, I had What if we know? Take the first row. It's attract the third row from and put that in the first row. So you one minus zero is 1 1/3 minus zero is 1/3 1 minus one is 00 minus seven houses negative. Seven halves 1/3 minus one is negative. 2/3 and zero zero minus negative. Three is three. Keep the other rose the same and I think just one more step would be needed here. What if we add negative 1/3 time's the second rose to the first row and replace the first road with that? So no, the first road becomes negative. 1/3 time. Zero close one Just one negative. 1/3 times one is negative 1/3 plus 1 30 0 Negative. 1/3 time. Zero 00 now negative 1/3 times. Three. It's negative. One plus negative. Seven. House His negative nine hands Negative 1/3 times one is negative. 1/3 plus negative. 2/3 is negative. One negative. 1/3 times negative three is one plus 34 and now we have the identity matrix on the left, therefore are inverse matrix is on the right and it is negative. Nine past negative. 14 31 negative three seven halves, one negative three


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