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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.$$\left[\begin{array}{rrr}{5} & {0} & {0} \\ {-1} & {1} & {0} \\ {-2} & {3} & {-1}\end{array}\right]$$

$A^{-1}=-\frac{1}{5}\left[\begin{array}{ccc}{-1} & {0} & {0} \\ {-1} & {-5} & {0} \\ {-1} & {-15} & {5}\end{array}\right]$

Algebra

Chapter 3

Determinants

Section 3

Cramer’s Rule, Volume, and Linear Transformations

Introduction to Matrices

McMaster University

Harvey Mudd College

Lectures

01:32

In mathematics, the absolu…

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07:40

In Exercises 11–16, comput…

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02:54

Find the inverse of each m…

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For the following exercise…

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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's calculate the determinants of a I'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'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'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's negative. 1/5 times matrix negative 100 Negative one negative 50 Negative one negative 15 5

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