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In Exercises 11–16, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.$$\left[\begin{array}{lll}{3} & {5} & {4} \\ {1} & {0} & {1} \\ {2} & {1} & {1}\end{array}\right]$$

$A^{-1}=\frac{1}{6}\left[\begin{array}{ccc}{-1} & {-1} & {5} \\ {1} & {-5} & {1} \\ {1} & {7} & {-5}\end{array}\right]$

Algebra

Chapter 3

Determinants

Section 3

Cramer’s Rule, Volume, and Linear Transformations

Introduction to Matrices

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University of Michigan - Ann Arbor

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were given a matrix and were asked to compute the aggregate of this matrix and to use the'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's find the determinant of this matrix. Do this by expanding across the second row, so we have the determinant. A. It'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'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

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