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Q17E

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Linear Algebra With Applications

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?role="math" localid="1659492178067" $B^{- 1}$ .

The Matrix is A + B 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, $A = A^{T}$ .

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

${(A + B)}^{T} = A^{T} + B^{T} = A + B$

Therefore, A + B is a symmetric matrix.

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Consider a basis ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m}$ of a subspace V of role="math" localid="1659434380505" $ℝ^{n}$ . Show that a vector $\vec{x}$ in role="math" localid="1659434402220" $ℝ^{n}$ is orthogonal to V if and only if $\vec{x}$ is orthogonal to all vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m}$ .

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