Given the partitioned matrix A = [ B]. B]₁ C D in which B and C are invertible. Which of the following gives the correct formula for A-¹?
Given the partitioned matrix A = [ B]. B]₁ C D in which B and C are invertible. Which of the following gives the correct formula for 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
Related questions
Question
![### Inverse of a Partitioned Matrix
Given the partitioned matrix
\[ A = \begin{bmatrix} 0 & B \\ C & D \end{bmatrix} \]
in which \( B \) and \( C \) are invertible. Which of the following gives the correct formula for \( A^{-1} \)?
### Answer Options:
- \[
\begin{bmatrix}
-C^{-1} D B^{-1} & C^{-1} \\
B^{-1} & 0
\end{bmatrix}
\]
- \[
\begin{bmatrix}
0 & B^{-1} \\
C^{-1} & -B^{-1} D C^{-1}
\end{bmatrix}
\]
- \[
\begin{bmatrix}
0 & C^{-1} \\
B^{-1} & -B^{-1} D C^{-1}
\end{bmatrix}
\]
- \[
\begin{bmatrix}
0 & -B \\
-C & -D
\end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc353cc0f-0c4d-4d5b-a3e6-e746dbb90f94%2F1fd2ecc9-2a0c-4871-9ce3-9ef15833e00f%2Fhchz4j_processed.png&w=3840&q=75)
Transcribed Image Text:### Inverse of a Partitioned Matrix
Given the partitioned matrix
\[ A = \begin{bmatrix} 0 & B \\ C & D \end{bmatrix} \]
in which \( B \) and \( C \) are invertible. Which of the following gives the correct formula for \( A^{-1} \)?
### Answer Options:
- \[
\begin{bmatrix}
-C^{-1} D B^{-1} & C^{-1} \\
B^{-1} & 0
\end{bmatrix}
\]
- \[
\begin{bmatrix}
0 & B^{-1} \\
C^{-1} & -B^{-1} D C^{-1}
\end{bmatrix}
\]
- \[
\begin{bmatrix}
0 & C^{-1} \\
B^{-1} & -B^{-1} D C^{-1}
\end{bmatrix}
\]
- \[
\begin{bmatrix}
0 & -B \\
-C & -D
\end{bmatrix}
\]
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