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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
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.
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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