Find the adjoint of the matrix A. Then use the adjoint to find the inverse of A (if possible). (If not possible, e -5 -6 -7 3 5 2 0 A = adj(A) = A-1 = -7 13 23 5L 1 -1 -3 5 11 3 -5 -7
Find the adjoint of the matrix A. Then use the adjoint to find the inverse of A (if possible). (If not possible, e -5 -6 -7 3 5 2 0 A = adj(A) = A-1 = -7 13 23 5L 1 -1 -3 5 11 3 -5 -7
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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![**Finding the Adjoint and Inverse of a Matrix**
To find the adjoint of the matrix **A** and use it to find the inverse of **A** (if possible), follow the steps illustrated below.
Given the matrix:
\[
A = \begin{bmatrix}
-5 & -6 & -7 \\
3 & 5 & 2 \\
0 & 1 & -1
\end{bmatrix}
\]
1. **Compute the Adjoint of A (adj(A)):**
\[
\text{adj}(A) = \begin{bmatrix}
-7 & -3 & 3 \\
13 & 5 & -5 \\
23 & 11 & -7
\end{bmatrix}
\]
2. **Verification:**
These values are placed into their appropriate positions within the matrix.
3. **Inverse of A (A\(^{-1}\)) Documentation:**
If the inverse exists, it can be computed using the formula:
\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]
Here, it indicates that there is an error (denoted by the red "X" symbol) in finding the inverse. This may be due to \(\text{det}(A) = 0\), indicating **A** is singular and not invertible.
\[
A^{-1} = \begin{bmatrix}
\phantom{-} & \phantom{-} & \phantom{-} \\
\phantom{-} & \phantom{-} & \phantom{-} \\
\phantom{-} & \phantom{-} & \phantom{-}
\end{bmatrix}
\]
**Graph/Diagram Explanation:**
- The matrix **A** and its elements were initially displayed.
- The adjoint matrix \(\text{adj}(A)\) is calculated showing individual elements filled in their correct places.
- The green arrows point downward, and to the right, indicating steps or movement to the next stage of the process.
- The red "X" indicates an error or the impossibility of calculating \(A^{-1}\).
To find the inverse, one needs to compute the determinant of **A** and see if it is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e1fa912-5e0c-4d6e-bcb6-bc3942e62084%2F439cef29-fac2-44f7-90ac-5091076a44f9%2Fvdderdf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Adjoint and Inverse of a Matrix**
To find the adjoint of the matrix **A** and use it to find the inverse of **A** (if possible), follow the steps illustrated below.
Given the matrix:
\[
A = \begin{bmatrix}
-5 & -6 & -7 \\
3 & 5 & 2 \\
0 & 1 & -1
\end{bmatrix}
\]
1. **Compute the Adjoint of A (adj(A)):**
\[
\text{adj}(A) = \begin{bmatrix}
-7 & -3 & 3 \\
13 & 5 & -5 \\
23 & 11 & -7
\end{bmatrix}
\]
2. **Verification:**
These values are placed into their appropriate positions within the matrix.
3. **Inverse of A (A\(^{-1}\)) Documentation:**
If the inverse exists, it can be computed using the formula:
\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]
Here, it indicates that there is an error (denoted by the red "X" symbol) in finding the inverse. This may be due to \(\text{det}(A) = 0\), indicating **A** is singular and not invertible.
\[
A^{-1} = \begin{bmatrix}
\phantom{-} & \phantom{-} & \phantom{-} \\
\phantom{-} & \phantom{-} & \phantom{-} \\
\phantom{-} & \phantom{-} & \phantom{-}
\end{bmatrix}
\]
**Graph/Diagram Explanation:**
- The matrix **A** and its elements were initially displayed.
- The adjoint matrix \(\text{adj}(A)\) is calculated showing individual elements filled in their correct places.
- The green arrows point downward, and to the right, indicating steps or movement to the next stage of the process.
- The red "X" indicates an error or the impossibility of calculating \(A^{-1}\).
To find the inverse, one needs to compute the determinant of **A** and see if it is
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