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Expert-verified Found in: Page 309 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # If all the entries of an invertible matrix A are integers, then the entries of ${{\mathbit{A}}}^{\mathbf{-}\mathbf{1}}$ must be integers as well.

The given statement is false.

See the step by step solution

## 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 “ $m×n$ .”

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

To find ${A}^{-1}$ :

For example,

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

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

${A}^{-1}=\left[\begin{array}{cc}\frac{-1}{2}& \frac{3}{2}\\ \frac{1}{2}& \frac{-1}{2}\end{array}\right]$

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

So, the given statement is false. ### Want to see more solutions like these? 