Prove that if A is an invertible n x n matrix, then adj A is an invertible matrix and -1 (adj A)-¹ 1 -A. det A
Prove that if A is an invertible n x n matrix, then adj A is an invertible matrix and -1 (adj A)-¹ 1 -A. det A
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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![### Educational Content on Matrix Theory
**Statement to Prove:**
Prove that if \( A \) is an invertible \( n \times n \) matrix, then \(\text{adj } A\) (the adjugate of \( A \)) is an invertible matrix. Furthermore, show that:
\[
(\text{adj } A)^{-1} = \frac{1}{\det A} A.
\]
**Explanation:**
If \( A \) is an invertible matrix, then its determinant, \(\det A\), is non-zero. The adjugate of \( A \), denoted as \(\text{adj } A\), is defined as the transpose of the cofactor matrix of \( A \).
To prove the given statement:
1. Recall that for an invertible matrix \( A \), the formula for the inverse is:
\[
A^{-1} = \frac{1}{\det A} \text{adj } A.
\]
2. Substitute \(\text{adj } A = A^{-1} \det A\) into the equation, we get:
\[
(\text{adj } A)^{-1} = \frac{1}{\det A} A.
\]
This shows that \(\text{adj } A\) is invertible when \( A \) is invertible and satisfies the equation above.
The understanding of these relationships is crucial in advanced linear algebra, particularly in solving systems of equations and computing matrix inverses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30f79184-3047-455a-b3d8-f2ad47623cdf%2Fd35eec19-8134-452a-99f4-f51347e69c6e%2Fe9cfled_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content on Matrix Theory
**Statement to Prove:**
Prove that if \( A \) is an invertible \( n \times n \) matrix, then \(\text{adj } A\) (the adjugate of \( A \)) is an invertible matrix. Furthermore, show that:
\[
(\text{adj } A)^{-1} = \frac{1}{\det A} A.
\]
**Explanation:**
If \( A \) is an invertible matrix, then its determinant, \(\det A\), is non-zero. The adjugate of \( A \), denoted as \(\text{adj } A\), is defined as the transpose of the cofactor matrix of \( A \).
To prove the given statement:
1. Recall that for an invertible matrix \( A \), the formula for the inverse is:
\[
A^{-1} = \frac{1}{\det A} \text{adj } A.
\]
2. Substitute \(\text{adj } A = A^{-1} \det A\) into the equation, we get:
\[
(\text{adj } A)^{-1} = \frac{1}{\det A} A.
\]
This shows that \(\text{adj } A\) is invertible when \( A \) is invertible and satisfies the equation above.
The understanding of these relationships is crucial in advanced linear algebra, particularly in solving systems of equations and computing matrix inverses.
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