12. Let A be a fixed n x n matrix and let S = {B e Mnxn|AB = BA}. Is S a subspace of Mnxn?
12. Let A be a fixed n x n matrix and let S = {B e Mnxn|AB = BA}. Is S a subspace of Mnxn?
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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How do I know if S is a subspace of M nxn?
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![**Matrix Subspaces and Commutativity**
**Problem Statement**
12. Let \( A \) be a fixed \( n \times n \) matrix and let
\[ S = \{ B \in M_{n \times n} \mid AB = BA \}. \]
**Question**
Is \( S \) a subspace of \( M_{n \times n} \)?
**Explanation**
In this problem, students are asked to consider the set \( S \), which consists of all \( n \times n \) matrices \( B \) that commute with a given \( n \times n \) matrix \( A \). The goal is to determine whether \( S \) is a subspace of the vector space of all \( n \times n \) matrices \( M_{n \times n} \).
A subspace must satisfy three primary conditions:
1. The zero vector (in this case, the zero matrix) must be in the set.
2. The set must be closed under vector addition.
3. The set must be closed under scalar multiplication.
To demonstrate whether \( S \) meets these criteria, one would typically follow these steps:
1. **Zero Vector Inclusion**: Check if the zero matrix is in \( S \).
2. **Closure Under Addition**: Assume \( B_1, B_2 \in S \) and verify whether \( B_1 + B_2 \in S \).
3. **Closure Under Scalar Multiplication**: Assume \( B \in S \) and \( c \) is a scalar, and verify whether \( cB \in S \).
This analysis will help in concluding whether \( S \) is indeed a subspace of \( M_{n \times n} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d548be9-b70c-4345-ad21-fb156d23b920%2Fbc8c2724-ad22-44fd-9d69-09ed686e963f%2F63rorhq.png&w=3840&q=75)
Transcribed Image Text:**Matrix Subspaces and Commutativity**
**Problem Statement**
12. Let \( A \) be a fixed \( n \times n \) matrix and let
\[ S = \{ B \in M_{n \times n} \mid AB = BA \}. \]
**Question**
Is \( S \) a subspace of \( M_{n \times n} \)?
**Explanation**
In this problem, students are asked to consider the set \( S \), which consists of all \( n \times n \) matrices \( B \) that commute with a given \( n \times n \) matrix \( A \). The goal is to determine whether \( S \) is a subspace of the vector space of all \( n \times n \) matrices \( M_{n \times n} \).
A subspace must satisfy three primary conditions:
1. The zero vector (in this case, the zero matrix) must be in the set.
2. The set must be closed under vector addition.
3. The set must be closed under scalar multiplication.
To demonstrate whether \( S \) meets these criteria, one would typically follow these steps:
1. **Zero Vector Inclusion**: Check if the zero matrix is in \( S \).
2. **Closure Under Addition**: Assume \( B_1, B_2 \in S \) and verify whether \( B_1 + B_2 \in S \).
3. **Closure Under Scalar Multiplication**: Assume \( B \in S \) and \( c \) is a scalar, and verify whether \( cB \in S \).
This analysis will help in concluding whether \( S \) is indeed a subspace of \( M_{n \times n} \).
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