Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

All nonzero symmetric matrices are invertible.

The given statement is false.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of a symmetric matrix.

Suppose A is a $n \times n$ matrix.

Then the matrix is said to be symmetric matrix, if $A^{T} = A$ .

A matrix is said to be invertible, if the determinant of the matrix is non-zero.

Step 2: Check whether the given statement is a true or false.

The given statement is all nonzero symmetric matrices are invertible.

For example.

Take, $A = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$

Then,

$A^{T} = {[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]}^{T} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] = A A^{T} = {[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]}^{T} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] = A$

So, A is a symmetric matrix, but it is not invertible, because det(A)=0.

This means, every symmetric matrix cannot be invertible.

Then the given statement is false.

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Linear Algebra With Applications

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Short Answer

All nonzero symmetric matrices are invertible.

Step by Step Solution

Step 1: Definition of a symmetric matrix.

Step 2: Check whether the given statement is a true or false.

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Linear Algebra With Applications

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Short Answer

All nonzero symmetric matrices are invertible.

Step by Step Solution

Step 1: Definition of a symmetric matrix.

Step 2: Check whether the given statement is a true or false.

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