Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q20E (page 233)

Question: If the $n 脳 n$ matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? ${AB}^{2} A$ .

Short Answer

Expert verified

The Matrix ${AB}^{2} A$ is symmetric.

Step by step solution

Definition of symmetric matrix.

A square matrix is symmetric matrix if $A B^{2} A$

Verification whether the given matrix is symmetric.

Given that A and B are symmetric matrices and B is invertible.

Since A is symmetric, then $A = A^{T}$ .

Then by properties of transpose, ${(A B)}^{T} = B^{T} A^{T}$ it gives,

$\begin{aligned} {({AB}^{2} A)}^{T} = (A B B A)^{T} \\ = [(A B) (B A)]^{T} \\ = (B A)^{T} (A B)^{T} \\ = (A^{T} B^{T}) (B^{T} A^{T}) \end{aligned}$

Further solve the above expression,

${({AB}^{2} A)}^{T} = (A B) (B A) = A (B B) A = {AB}^{2} A$

Therefore, ${AB}^{2} A$ is a symmetric matrix.

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91影视

Question: If the $n 脳 n$ matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? ${AB}^{2} A$ .

Short Answer

Step by step solution

Definition of symmetric matrix.

Verification whether the given matrix is symmetric.

One App. One Place for Learning.

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91影视

Question: If the n脳nmatrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?AB2A.

Short Answer

Step by step solution

Definition of symmetric matrix.

Verification whether the given matrix is symmetric.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Probability and Statistics

Discrete Mathematics

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Question: If the $n 脳 n$ matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? ${AB}^{2} A$ .