Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

If all the entries of an invertible matrix A are integers, then the entries of $A^{- 1}$ must be integers as well.

The given statement is false.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Matrix Definition

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “ m by n ” matrix, written “ $m \times n$ .”

Step 2: To check whether the given condition is true or false

To find $A^{- 1}$ :

For example,

$A = [\begin{matrix} 1 & 2 \\ 1 & 1 \end{matrix}]$ is an invertible matrix with all integer entries.

By using $A^{- 1}$ formula, however

$A^{- 1} = [\begin{matrix} \frac{- 1}{2} & \frac{3}{2} \\ \frac{1}{2} & \frac{- 1}{2} \end{matrix}]$

Therefore, the A and $A^{- 1}$ both are different.

So, the given statement is false.

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Linear Algebra With Applications

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Short Answer

If all the entries of an invertible matrix A are integers, then the entries of $A^{- 1}$ must be integers as well.

Step by Step Solution

Step 1: Matrix Definition

Step 2: To check whether the given condition is true or false

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Linear Algebra With Applications

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Short Answer

If all the entries of an invertible matrix A are integers, then the entries of A-1 must be integers as well.

Step by Step Solution

Step 1: Matrix Definition

Step 2: To check whether the given condition is true or false

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If all the entries of an invertible matrix A are integers, then the entries of $A^{- 1}$ must be integers as well.