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If an n X n matrix B with det (B) + 0, then for each vector b in R", Bx = b_has several solutions x = B-lb, where B-l is the inverse of B. True O False

If an n X n matrix B with det (B) + 0, then for each vector b in R", Bx = b_has several solutions x = B-lb, where B-l is the inverse of B. True O False

Advanced Engineering Mathematics
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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If an \( n \times n \) matrix \( B \) with \(\det(B) \neq 0\), then for each vector \( b \) in \(\mathbb{R}^n\), \( Bx = b \) has several solutions \( x = B^{-1}b \), where \( B^{-1} \) is the inverse of \( B \). - True - False
Transcribed Image Text:If an \( n \times n \) matrix \( B \) with \(\det(B) \neq 0\), then for each vector \( b \) in \(\mathbb{R}^n\), \( Bx = b \) has several solutions \( x = B^{-1}b \), where \( B^{-1} \) is the inverse of \( B \). - True - False
**Question:** If \( A \) is invertible, then elementary row operations that reduce \( A \) to the identity \( I_n \) also reduce \( A^{-1} \) to \( I_n \). - ○ True - ○ False **Explanation:** This question asks whether the same elementary row operations that can transform a matrix \( A \) into the identity matrix \( I_n \) can also transform its inverse \( A^{-1} \) into the identity matrix \( I_n \). Elementary row operations are basic operations used in linear algebra, including row swapping, row multiplication, and row addition, which don't change the solution set of a system.
Transcribed Image Text:**Question:** If \( A \) is invertible, then elementary row operations that reduce \( A \) to the identity \( I_n \) also reduce \( A^{-1} \) to \( I_n \). - ○ True - ○ False **Explanation:** This question asks whether the same elementary row operations that can transform a matrix \( A \) into the identity matrix \( I_n \) can also transform its inverse \( A^{-1} \) into the identity matrix \( I_n \). Elementary row operations are basic operations used in linear algebra, including row swapping, row multiplication, and row addition, which don't change the solution set of a system.
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