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Q26E

Expert-verifiedFound in: Page 309

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** ${{\mathit{A}}}^{\mathbf{-}\mathbf{1}}$ **must be integers as well.**

The given statement is 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.

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