in-exercises-1116-compute-the-adjugate-of-the-given-matrix-and-then-use-theorem-8-to-give-the-inve-6

Question

In Exercises 11–16, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.
$$
\left[\begin{array}{rrr}{1} & {2} & {4} \ {0} & {-3} & {1} \ {0} & {0} & {-2}\end{array}ight]
$$

Viswesh Uppalapati · Accepted Answer

in this video, we&#x27;re gonna be solving problem number 16 from section 3.3, which is based on Kramer&#x27;s Warren determinants. So here were given this initial, uh, matrix A, and we&#x27;re told to find the inverse of a using their. The equation given interior meet with Basic says that the inverse of a matrix A is just one over. It&#x27;s a determinant, times the adjective of a And here the idea of a is listed because the computation of the Advocate is a long and tedious and will not be shown in this video because it would make the video too long. So to find the adjective A all you have to do is find the co factor matrix of A and transpose it. So to find a co factor of one element in a So let&#x27;s pick the 1st 1 you block out the row and call them. That is, um, that contains that element and take the two by two resulting determined of whatever&#x27;s left, which is one singular co packer. And you repeat, repeat that with every term in a and you basically transpose it, which will give you this matrix. Uh, next up is to find the determinant of A, which is one times native, three times negative, two minus one time, zero minus two times zero times negative. Too minus one time. Zero plus four times These are times or ah, minus morning news three times zero. Which gives us, um, meticulous six. So a inverse is just 1/6 time&#x27;s the address. So you just seal every element in the Advocate by 1/6. So you should get and rose to be one to third. Uh, 7/3 zero negative. 1/3. I got a born six zeros there on negative one. Now you get one, which is the inverse of the initial.

Viswesh Uppalapati · Answer

in this video, we&#x27;re gonna be going over Problem number Ah. 14 of section group went through, which is on Cramer&#x27;s rule in determinants. Uh, this problem gives us this matrix initial matrix and tells us to use the dirham. Are the formula government there, mate? Which is to find an inverse of matrix. You just take just scale the advocate of that matrix by one over the chairman of the initial matrix. Um, so and it tells us tow Use that to find the inverse of the matrix. So first we find the advocate of the Matrix, which I wrote here because it is very time consuming to do on video. But it is basically, um it is basically that transposed of the of every co far the cool factor matrix of a. So the cool factor of a transfers to find the cool factor. Cool factor is basically just the determinant, um, is basis a regular determinant of a two by two. So to find the cool factor of each term. So, for example, if you want to find the collector of this one, we basically block out this one on block out. Uh, this column. So the whatever row and call him that. That number is in which we&#x27;re trying to find the cool factor of the block goes out and find the determined of the resulting two by two that is left unlocked. Um, in case of a three by three. So you do that for every single term, and u transpose that, um, matrix and you&#x27;ll end up getting this. So next we&#x27;ll find the determinant of a ah, which is one times two times four minus one time, zero minus negative one times, uh, zero times for minus one times two and plus two times zero time zero minus two times two. Which gives us determined of a to B negative, too. It should evaluate a negative on DDE. We can find the inverse easily by, um, Skilling every, uh, term. And as you get a bye negative 1/2 So I should just give us negative 4 92 five times negative One is your uh huh. And to one negative war, which is the inverse of the of makers say we&#x27;re here

Viswesh Uppalapati · Answer

in this video, we&#x27;re gonna be solving problem number 12 of section 3.3, which is on determinants and, uh, determinants and Cramer&#x27;s rule. So this problem gives us ah, matrix A and it tells us tow use theory. The equation given wasn&#x27;t there, mate, which is the inverse of a equals one over the determinant of the initial matrix times, the advocate of the Matrix, which is given right here. So I previously calculated that as you get of this matrix and already had it while the problems started because it would be time consuming for the video. But I&#x27;ll show you howto Ah, take the adjective of matrix. It&#x27;s an adjective. A matrix is basically the transpose of the co factors of every term in the Matrix. So it&#x27;s just co factor of a transposed. So how do you find a co factor is basically, um, you take every possible determinant in this three by three matrix right here by blocking out the road that that element is in. So if you want to take a co factor of this one, let&#x27;s say we ignore the first true and take the determinant of the, uh the, uh these 21 and 11 terms. If you want to find the determinant of ako factor of this one, we ignore that row and find the determinant of these two. And if you want to take the 3rd 1 then we ignore these two arose and take the determinant of Is tha these two columns Sorry for being a mess. And you repeat that with every single every single row in the Matrix and U transpose it u transpose the result into co factor matrix, um, to get the advocate of the Matrix. So to use this formula, we need to first find the determinant of A, which is basically like finding Kovac also one of the poor. In fact, and finding a cool factor is, um, every co factor is alternating signs. So the 1st 1 starts with a plus the next one&#x27;s minus plus minus, plus minus, plus minus a plus. Um, so moving on to find the determinant of a, um is the same thing so that here it would be one times two times on minus one times one minus one times two times one minus one times zero Look, plus three times. Um, here It&#x27;s a negative. Too negative, too. Times one miners one times are too times around. And after we simplify, we should get one times warming minus plus two. Hold on one sec. I&#x27;m sorry. I should be. Should be, um one times two minus negative. So that would be a plus two and then, uh, minus six. It&#x27;s t minus one. So this is actually one. I&#x27;m sorry. Look, And minus three R money six. So you should get three minus six equals negative. Three. So that&#x27;s the initial determinant. So a inverse is just 10 negative. 1/3 times. Um, 12 negative. Five to one. Negative seven. Negative to a negative one for. So this means that we scale every factor or every term in dodge. A bit of a with by negative negative to me. So it&#x27;s just negative one there times it you to these numbers in the resulting matrix

Chris Trentman · Answer

we&#x27;re giving him Beatrix and whereas to compute the aggregate of the Given Matrix and to use their, um eight to find the inverse of the Matrix. Yeah, The Matrix is thief three by three matrix zero negative to negative. One 500 Negative. 111 We&#x27;ll call this matrix a. So to find the advocate of this matrix. First, let&#x27;s find the determinant. So expanding across thes second row so that we get negative five times the determinant the matrix negative to one negative one 11 says negative five times negative two plus one or negative one just simply five. And then let&#x27;s calculate the co factors of a So we see that see 11 is the determinant of the matrix 0011 which is zero c two c 12 is going to be the opposite of the determinant of the matrix. 50 Negative 11 This is the opposite of five is negative five and next we calculate See 13 This is determinant of the matrix. 50 negative 11 c 21 is theocracies it of the determinant of the matrix. Negative to negative. 111 and as we calculated before, this is negative. Negative one which is a positive one. C 22 is the determinant matrix zero negative one negative 11 This is negative. One C 23 is the opposite of the determinant. Zero negative to negative 11 This is the opposite of negative too, which is simply too. C 31 is the determinant of the matrix negative to negative 100 which is zero C 32 is theocracies it of the determinants of zero negative 150 This is the opposite of 65 which is negative. Five finally see 33 is the determinant the matrix zero negative to 50 This is 10. So using the co factors, we can construct the matrix of co factors. Use this to form the ad joint. Well, we have that The ad joint of a I guess it&#x27;s the ad you get for the classical Add joint Obey. This is the matrix with entries. Well, we have C 11 get for left corner. She zero. Then we have seen to one which is one and then see 31 which is zero and next role. We start off with C 12 which is negative five, followed by C 22 which is negative one, followed by C 32 which is negative. Five. In the next room, we have see 13 which is five c 23 which is to, and C 33 which is 10. So this is the aggregate of this matrix. And by your, um eight we have at the inverse of this Matrix aid is one over the determinant. Okay, times the educated A. This is equal to recall that we calculated the determinant to be five. This is one assist of the matrix with entries 010 negative five negative or negative five 5 to 10.

Chris Trentman · Answer

were given a matrix and rest to compute the aggregate of this matrix and to use their eight to find the inverse of this matrix exists. The Matrix is a three by three matrix a centuries 500 negative 110 negative 23 Negative one. First, that&#x27;s calculate the determinants of a I&#x27;ll do this by expanding across the top row so we have five times the determinant of sub matrix. 10 three Negative one since five times you have one which is negative. Five. Next, let&#x27;s calculate the co factors. We have C 11 This is, as we had earlier, the determinant of the sub metrics. 10 three negative one. This is simply negative one c 12 which is the opposite of the determinant of subject tricks. Negative 10 negative to negative. One says the opposite of one. Just negative one. Next. See 13 This is determinant of the sub matrix Negative 11 negative 23 it says negative three plus two, which is negative. One c 21 This is the opposite of the determinant of the sub matrix 003 Negative one. This is simply zero c 22 which is the determinant of the sub matrix 50 to negative one, which is negative. Five c 23 which is the opposite of the determinant of the sub matrix 50 negative 23 which is negative. 15 c 31 which is the determined of the sub metrics 0010 which is zero c 32 which is the opposite of the determinant of these sub matrix 50 negative 10 which is zero and C 33 which is the determinant of the sub matrix 50 negative 11 which is five. And we have that The aggregate of A is defined to be the transpose of the matrix of co factors. So for the first row, the first column we have C 11 c one to C 13 or negative one negative one negative one. In the next column, we have C 21 which is zero, followed by C 22 which is negative five, followed by seat to three, which is negative. 15. You&#x27;re crazy. And in the last column we have C 31 which is zero, followed by C 32 which is zero, followed by C 33 which is five. So this is the advocate. They and by theory, mates we have that The inverse today is one over the determinant. Today times the educative a recall that we calculated the determinant today was negative five. So it follows that the inverse today it&#x27;s negative. 1/5 times matrix negative 100 Negative one negative 50 Negative one negative 15 5

Chris Trentman · Answer

were given a matrix and were asked to compute the aggregate of this matrix and to use the&#x27;re, um eight to find the inverse of this matrix if it exists. So the Matrix is the three by three Matrix A which has entries 354 101 211 So first, let&#x27;s find the determinant of this matrix. Do this by expanding across the second row, so we have the determinant. A. It&#x27;s going to be negative one times to determine into the matrix 5411 minus one times the determinant of the matrix. 35 to 1. And so we get negative one times five minus four, which is negative one times one which is simply negative. One minus one times three minus 10. So negative one times negative seven or just positive seven. So they determined to six. Now let&#x27;s calculate the co factors we have See 11 is the determinant of the sub matrix 0111 which is negative. One c 12 is the determinant of the sub matrix. 1121 This is one minus two or negative one. Actually, this should be the opposite of this determinants that should be positive. See, 13 is the determinant of these sub matrix 0 to 1. This is one C 21 is the opposite of the determinant of the sub matrix 54 on one which is negative. One c 22 is the determined of the sub matrix 34 to 1. This is three minus eight, which is negative. Five C 23 is thes opposite of the determinant of the sub matrix 35 21 This is the opposite of negative seven, which is positive. Seven c 33 c 31 I mean is the determinant of the sub matrix 5401 This is five C 32 is the opposite of the determinant of the sub metrics 3411 the opposite of negative one which is simply one. Finally see 33 is the determinant of the sub matrix 3510 which is negative five. And so with all these co factors, you can find the educative a which is the transpose of the matrix of co factors. So this has three entries very much, see 11 which is a negative one and then see 21 which is negative. One all of I C 31 which is five. The next row, starting with C 12 which is one followed by C 22 which is negative five, followed by C 32 which is one. And the next room starting off with C 13 which is one followed by C 23 which is seven, followed by C 33 which is negative thoughts. And so this is the aggregate of a and by theorem eight. We have that The inverse today is one over the determinant. Today times the advocate of a So call it we calculated the determinant was six. So this is 1/6 of the matrix. Negative one negative. 15 one negative 51 17 negative five

Problem

Show that if $A$ is $2 \times 2,$ then Theorem 8 …

Problem 16 Medium Difficulty

In Exercises 11–16, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.
$$
\left[\begin{array}{rrr}{1} & {2} & {4} \\ {0} & {-3} & {1} \\ {0} & {0} & {-2}\end{array}\right]
$$

Answer

$A^{-1}=\frac{1}{6}\left[\begin{array}{ccc}{6} & {4} & {14} \\ {0} & {-2} & {-1} \\ {0} & {0} & {-3}\end{array}\right]$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Kayleah T.

Caleb E.

Maria G.

Kristen K.

Absolute Value - Example 1

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$A^{-1}=\frac{1}{6}\left[\begin{array}{ccc}{6} & {4} & {14} \\ {0} & {-2} & {-1} \\ {0} & {0} & {-3}\end{array}\right]$

Linear Algebra and Its Applications

Kayleah T.

Caleb E.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Kayleah T.

Caleb E.

Maria G.

Kristen K.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications