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Q16E

Expert-verified

Found in: Page 233

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Question: If the $n \times n$ matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?A+B.

The Matrix A + B is symmetric.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of symmetric matrix.

A square matrix is symmetric matrix if $A = A^{T}$

Step 2: 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, ${(A + B)}^{T} = A^{T} + B^{T} = A + B$ it gives,

Therefore, A + B is a symmetric matrix.

Most popular questions for Math Textbooks

Consider a linear transformation L from $R^{m}$ to $R^{n}$ that preserves length. What can you say about the kernel of L? What is the dimension of the image? What can you say about the relationship between n and m? If A is the matrix of L, What can you say about the columns of A? What is $A^{T} A$ ? What about ${AA}^{T}$ ? Illustrate your answer with an example where m=2 and n=3.

a. Give an example of a (nonzero) skew-symmetric 3 × 3 matrix A, and compute.

b. If an n × n matrix A is skew-symmetric, is matrix $A^{2}$ necessarily skew-symmetric as well? Or is $A^{2}$ necessarily symmetric?

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

$4 \cdot [\begin{matrix} 4 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 25 \\ 0 \\ - 25 \end{matrix}], [\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}]$

Question: Consider an $n \times m$ matrix A, a vector in $R^{m}$ , and a vector $\vec{w}$ in $R^{m}$ . Show that $(A \vec{v}) . \vec{w} = \vec{v} . (A^{T} \vec{w})$ .

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

15. $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$

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Linear Algebra With Applications

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Short Answer

Question: If the $n \times n$ matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?A+B.

Step by Step Solution

Step 1: Definition of symmetric matrix.

Step 2: Verification whether the given matrix is symmetric.

Most popular questions for Math Textbooks

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Linear Algebra With Applications

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Short Answer

Question: If the n×n matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?A+B.

Step by Step Solution

Step 1: Definition of symmetric matrix.

Step 2: Verification whether the given matrix is symmetric.

Most popular questions for Math Textbooks

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Question: If the $n \times n$ matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?A+B.