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Find the adjoint of the matrix A_ Then use the adjoint to find the inverse of A (if possible). (If not possible, enter IMPOSSIBLE:)A =adj(A)A-1 ,...

Question

Find the adjoint of the matrix A_ Then use the adjoint to find the inverse of A (if possible). (If not possible, enter IMPOSSIBLE:)A =adj(A)A-1 ,

Find the adjoint of the matrix A_ Then use the adjoint to find the inverse of A (if possible). (If not possible, enter IMPOSSIBLE:) A = adj(A) A-1 ,



Answers

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

What the determined matrix given on the left here is convertible and if it is fine it's in verse eight of the negative. First by the ad joint method. So let's no it's an important definitions. First of all, the inverse of a is equal to buy the ad joint method one over the determinative eight times the joint today where the ad join today is the transpose of a matrix formed by the co factors for each entry and A. And we have to make a note of the fact that A. Is in veritable if it's determined does not equal zero. So first let's find the determinant of a to make sure it can be inverted determinant of A is simply shoot times 93 minutes, zero minus 5,000,003 minutes. Zero plus five to negative four plus two or negative one. This is not equal zero. So we can invert a Let's now find its inverse by the joint method. The co factor matrix for a. I. J. Is given on the left here, and simply taking the transpose gives us the matrix on the right next to find the inverse of A. We simply have to multiply every single one of these entries by one over determining A or one over negative one, which gives inverse A. As is boxed in at the bottom here.

As I asked him were given a matrix and resting Slagle it of this matrix is convertible and if so to use the add joint method Find its inverse. This is the three x 3 matrix a equals 255 other A -1 -10 and 243 shitty faggot. That's me first is finally determined into this matrix A. This is the determinant of this three x 3 matrix. Using co factor expansions will expand across the top row. This is two times the determinant negative 1043 a minus five times the determinant of negative 10 to 3 plus five times the determinant of negative one. Negative 1 to 4. Down this matter this is equal to two times negative three minus zero minus five times negative three minus zero plus five times negative four plus two. Which simplifies to negative six plus 15 minus 10. Which is equal to negative one. Which of course is non zero Since are determined as non zero. It's it follows that our matrix A is in fact in vertebral. I I fear him from this section. Remember told somebody Now to find its inverse first. Let's find co factors of a. So the co factors C11 is the determinant of -1043. Of course, it's not true. Okay. Damn. Which is -3. co fracture. See 1 2. This is the opposite of the determinant of -10 23 Actually his second stuff three. and the cool factor C13. This is the determinant of the matrix negative one. Negative 1 to 4. Right. Right, right. This is negative four minus negative two. Or negative to co factor C 21 This is the opposite The determinant 5543. This is the opposite of 15 -20 or positive five. I'm stuck. The co factor C 22 This is the Derivative after this is the determinant of 2 5-3 lost. And this is negative four is the big guy. Co factors C 23 is the opposite of the determined 2, 5- four. He didn't bad news, which is to the future. I'm sorry mm. Likewise, we find that the co factor C31 is five. Co factors C32 is negative five. And the co factor C 33 is three. My turn. Therefore, with all these co factors, you can form a matrix of co factors. So any woman. Yeah. And so this is the matrix The country's negative 3 3 -2, 5 -4, 2, jewish man And 5 -5. 3. Come on, that's me, 14 year old Adam Friedland. And therefore the ad joint matrix. Perhaps the classical adjunct matrix hard to say with action means at join today. Well this is defined to be the transposed of the matrix of co factors of A. And so you're looking at our matrix of co factors. This is the matrix negative 355 Then I have three negative two, negative four negative 52 And number three like this. Now we know from the theorem at a inverse is equal to one over the determinant of the matrix A Times the ad joined today. Take you to get new since the determinant was negative one. This gives us the matrix centuries of the opposite of the edge. So we have positive three negative five, negative five, negative three, positive four, positive five And positive to -2 -3. And so this is our inverse, a inverse.

For matrix A given on the left. We want to determine if A. Is in veritable and if it is, we want to find the inverse matrix eight of the negative. First. Using the add joint method, we have to make note of a couple important definitions in order to solve this. So to start remember that the inverse matrix as found by the joint method is one over the determinant of age and the joint debate where the joint of A. It's just the transpose of a matrix formed by the co factors of each country and A. We also have to know the second definition here. A is inevitable if and only if the determinant does not equal zero. So for matrix A as we've noted already in the left, let's start to finance determinant for three by three matrix we use the formula german A equals two times six minutes, zero minus 0 to 48 minus zero plus 2 to 24 plus five or 12. Since this is not zero is inevitable and we receive defining a negative one. So first the co factor matrix is given here on the left and then by taking the transpose of the co factor matrix, we obtain the matrix on the right now that we have the transposed co factor matrix or the ad joint matrix. For a. We simply divide every entry by the determinant to obtain the inverse matrix. A negative one. Eyes is given here.

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.


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