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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Question
Why did I get -1 when it should be 1 for the inverse of A? what did i do wrong?
![**Matrix Inversion Example**
**Objective:**
Determine the inverse of matrix \(A\).
**Given:**
\[ A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 1 & 0 & 6 \end{bmatrix} \]
**Steps:**
1. **Calculate the Determinant of \(A\):**
- Find determinant \(|A|\):
\[
|A| = 6(-6) = -36
\]
- Simplified determinant:
\[
|A| = 6
\]
2. **Form the Matrix \(N\):**
- Simplified form:
\[
N = \begin{bmatrix} 0 & -2 & -6 \\ 3 & 0 & -3 \\ 0 & 2 & 0 \end{bmatrix}
\]
3. **Further Simplify Matrix \(N\):**
- Switch rows for determinant calculation:
\[
N = \begin{bmatrix} 0 & -3 & 0 \\ 2 & 0 & -2 \\ 6 & 3 & 0 \end{bmatrix}
\]
- Reduction and simplification:
\[
N = \begin{bmatrix} 0 & 3 & 0 \\ -2 & 0 & 2 \\ -6 & -3 & 0 \end{bmatrix}
\]
4. **Compute the Inverse of \(A\):**
- Formula:
\[
A^{-1} = \frac{1}{|A|}[N]
\]
- Calculation:
\[
A^{-1} = \frac{1}{6} \begin{bmatrix} 0 & -2 & -6 \\ 3 & 0 & -3 \\ 0 & 2 & 0 \end{bmatrix}
\]
- Result:
\[
A^{-1} = \begin{bmatrix} 0 & -1/3 & 0 \\ 1/2 & 0 & -1/2 \\ 0 & 1/3 & 0 \end{bmatrix}
\]
**Conclusion:**
- The inverse of matrix \(A\) is \(\begin{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeb06b0e-8341-4861-846e-a0416daf2b6f%2F700e0e70-1e6a-415f-b2da-bc6bfe090ffc%2Fnml51lv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Inversion Example**
**Objective:**
Determine the inverse of matrix \(A\).
**Given:**
\[ A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 1 & 0 & 6 \end{bmatrix} \]
**Steps:**
1. **Calculate the Determinant of \(A\):**
- Find determinant \(|A|\):
\[
|A| = 6(-6) = -36
\]
- Simplified determinant:
\[
|A| = 6
\]
2. **Form the Matrix \(N\):**
- Simplified form:
\[
N = \begin{bmatrix} 0 & -2 & -6 \\ 3 & 0 & -3 \\ 0 & 2 & 0 \end{bmatrix}
\]
3. **Further Simplify Matrix \(N\):**
- Switch rows for determinant calculation:
\[
N = \begin{bmatrix} 0 & -3 & 0 \\ 2 & 0 & -2 \\ 6 & 3 & 0 \end{bmatrix}
\]
- Reduction and simplification:
\[
N = \begin{bmatrix} 0 & 3 & 0 \\ -2 & 0 & 2 \\ -6 & -3 & 0 \end{bmatrix}
\]
4. **Compute the Inverse of \(A\):**
- Formula:
\[
A^{-1} = \frac{1}{|A|}[N]
\]
- Calculation:
\[
A^{-1} = \frac{1}{6} \begin{bmatrix} 0 & -2 & -6 \\ 3 & 0 & -3 \\ 0 & 2 & 0 \end{bmatrix}
\]
- Result:
\[
A^{-1} = \begin{bmatrix} 0 & -1/3 & 0 \\ 1/2 & 0 & -1/2 \\ 0 & 1/3 & 0 \end{bmatrix}
\]
**Conclusion:**
- The inverse of matrix \(A\) is \(\begin{
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