If the n x n matrices A and B are symmetric and B is in- vertible, which of the matrices in Exercises 13 through 20 must be symmetric as well? 13. ЗА 14. — В 15. АВ 16. А + В 17. В -1 18. A10 19. 21, + ЗА —4A2 20. АВ?А |
If the n x n matrices A and B are symmetric and B is in- vertible, which of the matrices in Exercises 13 through 20 must be symmetric as well? 13. ЗА 14. — В 15. АВ 16. А + В 17. В -1 18. A10 19. 21, + ЗА —4A2 20. АВ?А |
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 Exercise: Symmetry in Matrix Operations**
*Context:*
For the square matrices \( n \times n \), where matrices \( A \) and \( B \) are given to be symmetric and \( B \) is invertible, determine which of the matrices in the following exercises must also be symmetric.
**Exercises:**
13. \( 3A \)
14. \( -B \)
15. \( AB \)
16. \( A + B \)
17. \( B^{-1} \)
18. \( A^{10} \)
19. \( 2I_n + 3A - 4A^2 \)
20. \( AB^2A \)
*Note:*
- A matrix \( A \) is symmetric if \( A = A^T \).
- The identity matrix \( I_n \) is symmetric.
- The product of symmetric matrices is not necessarily symmetric unless they commute \( (AB = BA) \).
**Analysis:**
- Exercise 13 considers the scalar multiplication of \( A \), which preserves symmetry.
- Exercise 14 involves the negation of \( B \), which also preserves symmetry.
- Exercise 15 and 20 involve matrix multiplication, which depends on the commutative property.
- Exercise 16 involves the sum of two symmetric matrices.
- Exercises 17 deals with the inverse of a symmetric matrix.
- Exercise 18 involves raising a symmetric matrix to a power.
- Exercise 19 involves a linear combination of symmetric matrices and uses the identity matrix.
Use this guide to explore symmetric properties in these particular operations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce768285-628a-4724-9f2b-b5d3bc5907ae%2Fe2629a2c-8fc3-42c2-a335-39fc3bcae780%2Fbps3y1_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Exercise: Symmetry in Matrix Operations**
*Context:*
For the square matrices \( n \times n \), where matrices \( A \) and \( B \) are given to be symmetric and \( B \) is invertible, determine which of the matrices in the following exercises must also be symmetric.
**Exercises:**
13. \( 3A \)
14. \( -B \)
15. \( AB \)
16. \( A + B \)
17. \( B^{-1} \)
18. \( A^{10} \)
19. \( 2I_n + 3A - 4A^2 \)
20. \( AB^2A \)
*Note:*
- A matrix \( A \) is symmetric if \( A = A^T \).
- The identity matrix \( I_n \) is symmetric.
- The product of symmetric matrices is not necessarily symmetric unless they commute \( (AB = BA) \).
**Analysis:**
- Exercise 13 considers the scalar multiplication of \( A \), which preserves symmetry.
- Exercise 14 involves the negation of \( B \), which also preserves symmetry.
- Exercise 15 and 20 involve matrix multiplication, which depends on the commutative property.
- Exercise 16 involves the sum of two symmetric matrices.
- Exercises 17 deals with the inverse of a symmetric matrix.
- Exercise 18 involves raising a symmetric matrix to a power.
- Exercise 19 involves a linear combination of symmetric matrices and uses the identity matrix.
Use this guide to explore symmetric properties in these particular operations.
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