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Q14E

Expert-verified

Found in: Page 233

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

If the nxn matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -B.

The Matrix -B is symmetric.

See the step by step solution

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, $(kA) = {kA}^{T}$ it gives,

${(- B)}^{T} = - B^{T} = - B$

Therefore, -B is a symmetric matrix.

Most popular questions for Math Textbooks

Find the least square ${\overset{⇀}{x}}^{*}$ of the system $A \overset{⇀}{x} = b$ where $A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]$ and $\overset{⇀}{b} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$ .

Leonardo da Vinci and the resolution of forces. Leonardo (1452–1519) asked himself how the weight of a body, supported by two strings of different length, is apportioned between the two strings.

Three forces are acting at the point D: the tensions $F_{1}$ and $F_{2}$ in the strings and the weight $\vec{W}$ . Leonardo believed that

$\frac{| | {\vec{F}}_{1} | |}{| | {\vec{F}}_{2} | |} = \frac{\bar{E} \bar{A}}{\bar{E} \bar{B}}$

Was he right? (Source: Les Manuscripts de Léonard de Vinci, published by Ravaisson-Mollien, Paris, 1890.)

Hint: Resolve $F_{1}$ into a horizontal and a vertical component; do the same for $F_{2}$ . Since the system is at rest, the equation ${\vec{F}}_{1} + {\vec{F}}_{2} + \vec{W} = \vec{0}$ holds. Express the ratios

$\frac{| | {\vec{F}}_{1} | |}{| | {\vec{F}}_{2} | |}$ and $\frac{\bar{E} \bar{A}}{\bar{E} \bar{B}}$ . In terms of $α$ and $β$ , using trigonometric functions, and compare the results.

Consider an invertible n × n matrix A. Can you write A = RQ, where R is an upper triangular matrix and Q is orthogonal?

In Exercises 40 through 46, consider vectors $v ⃗_{1}, v ⃗_{2}, v ⃗_{3}$ in $R^{4}$ ; we are told that $v ⃗_{i}, v ⃗_{j}$ is the entry $a_{i j}$ of matrix A.

$A = [\begin{matrix} 3 & 5 & 11 \\ 5 & 9 & 20 \\ 11 & 20 & 49 \end{matrix}]$

Find a nonzero vector $\overset{⇀}{v}$ in span $({\overset{⇀}{v}}_{2} {\overset{⇀}{v}}_{3})$ such that $\overset{⇀}{v}$ is orthogonal to ${\overset{⇀}{v}}_{3}$ .Express as a linear combination of localid="1659441496004" ${\overset{⇀}{v}}_{2}$ and ${\overset{⇀}{v}}_{3}$ .

Find the angle $θ$ between each of the pairs of vectors $\overset{⇀}{u}$ and $\overset{⇀}{v}$ in exercises 4 through 6.

4. $\overset{⇀}{u} = [\begin{matrix} 1 \\ 1 \end{matrix}], \overset{⇀}{v} = [\begin{matrix} 7 \\ 11 \end{matrix}] .$

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Linear Algebra With Applications

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Short Answer

If the nxn matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -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

If the nxn matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -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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