Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Exercise 7.6.5**
Use the adjugate formula to find the inverse of the matrix
\[ A = \begin{bmatrix} 1 & 1 & 0 \\ 3 & 1 & 2 \\ 2 & -2 & 5 \end{bmatrix} \]
---
In this exercise, you are tasked with finding the inverse of a 3x3 matrix using the adjugate formula. A matrix \( A \) can be inverted by taking the transpose of its cofactor matrix (also known as the adjugate), and dividing each term by the determinant of \( A \).
1. **Cofactor Matrix**: Calculate the cofactor of each element in matrix \( A \).
2. **Adjugate (Adjoint) Matrix**: Transpose the cofactor matrix.
3. **Determinant**: Compute the determinant of matrix \( A \).
4. **Inverse Formula**: Use the formula \( A^{-1} = \frac{1}{\text{det}(A)} \times \text{Adj}(A) \).
This method allows the computation of the inverse, provided the determinant is non-zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98a85d3f-1221-4cd9-b5c2-c225eed9a46f%2F10a49193-c24c-4ab4-bb94-a0172b30ff10%2Fzff0dlf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 7.6.5**
Use the adjugate formula to find the inverse of the matrix
\[ A = \begin{bmatrix} 1 & 1 & 0 \\ 3 & 1 & 2 \\ 2 & -2 & 5 \end{bmatrix} \]
---
In this exercise, you are tasked with finding the inverse of a 3x3 matrix using the adjugate formula. A matrix \( A \) can be inverted by taking the transpose of its cofactor matrix (also known as the adjugate), and dividing each term by the determinant of \( A \).
1. **Cofactor Matrix**: Calculate the cofactor of each element in matrix \( A \).
2. **Adjugate (Adjoint) Matrix**: Transpose the cofactor matrix.
3. **Determinant**: Compute the determinant of matrix \( A \).
4. **Inverse Formula**: Use the formula \( A^{-1} = \frac{1}{\text{det}(A)} \times \text{Adj}(A) \).
This method allows the computation of the inverse, provided the determinant is non-zero.
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