Find the adjoint of the matrix A. Then use the adjoint to find the inverse of A (if possible). (If not possible, enter IMPOSSIBLE.) 1 3 A = 2 4 |4 -3 adj(A) |-2 1 A-1 =
Find the adjoint of the matrix A. Then use the adjoint to find the inverse of A (if possible). (If not possible, enter IMPOSSIBLE.) 1 3 A = 2 4 |4 -3 adj(A) |-2 1 A-1 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Exercise: Finding the Adjoint and Inverse of a Matrix**
Objective: Find the adjoint of the matrix \( A \). Then use the adjoint to find the inverse of \( A \) (if possible). If not possible, enter "IMPOSSIBLE."
**Given Matrix:**
\[
A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}
\]
**Solution Steps:**
1. **Adjoint of Matrix \( A \):**
- The adjoint of matrix \( A \) is obtained by taking the transpose of the cofactor matrix of \( A \).
\[
\text{adj}(A) = \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix}
\]
- The first element (4) is obtained by swapping the elements on the main diagonal of \( A \).
- The off-diagonal elements are negated (-3 and -2).
2. **Inverse of Matrix \( A \):**
- The inverse \( A^{-1} \) can be found using the formula:
\[
A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A)
\]
- Determine if \( \det(A) \neq 0 \) for \( A^{-1} \) to exist.
- Substitute the values to find the inverse matrix.
**Note:** The checkmark indicates that all steps have been verified, ensuring the possibility to calculate \( A^{-1} \). If \( \det(A) = 0 \), then the inverse is "IMPOSSIBLE."
---
This exercise demonstrates calculating the adjoint and verifying the computation of an inverse matrix for educational purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc42b80bf-a5d4-414b-bce1-0fe52a04dbbd%2Fbce90b95-6e30-44d1-847a-5b3ccf517dcb%2Fd8hidn_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise: Finding the Adjoint and Inverse of a Matrix**
Objective: Find the adjoint of the matrix \( A \). Then use the adjoint to find the inverse of \( A \) (if possible). If not possible, enter "IMPOSSIBLE."
**Given Matrix:**
\[
A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}
\]
**Solution Steps:**
1. **Adjoint of Matrix \( A \):**
- The adjoint of matrix \( A \) is obtained by taking the transpose of the cofactor matrix of \( A \).
\[
\text{adj}(A) = \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix}
\]
- The first element (4) is obtained by swapping the elements on the main diagonal of \( A \).
- The off-diagonal elements are negated (-3 and -2).
2. **Inverse of Matrix \( A \):**
- The inverse \( A^{-1} \) can be found using the formula:
\[
A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A)
\]
- Determine if \( \det(A) \neq 0 \) for \( A^{-1} \) to exist.
- Substitute the values to find the inverse matrix.
**Note:** The checkmark indicates that all steps have been verified, ensuring the possibility to calculate \( A^{-1} \). If \( \det(A) = 0 \), then the inverse is "IMPOSSIBLE."
---
This exercise demonstrates calculating the adjoint and verifying the computation of an inverse matrix for educational purposes.
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