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Q18E

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Linear Algebra and its Applications
Found in: Page 165
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Question: 18. Suppose that all the entries in A are integers and \({\bf{det}}\,A = {\bf{1}}\). Explain why all the entries in \({A^{ - {\bf{1}}}}\) are integers.

Here, all the entries in \({A^{ - 1}}\) are integers. This is because all entries in the adjugate matrix of A are integers.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Describe the given statement

Given, all entries in A are integers and \(\det A = 1\).

Step 2: Use the formula for cofactors

The cofactor of A is:

\({C_{ij}} = {\left( { - 1} \right)^{i + j}}{A_{ij}}\)

Since \({A_{ij}}\) is the sum of the product of entries in A, \({A_{ij}}\) is an integer. Thus \({C_{ij}}\) is an integer.

Hence, all the cofactors of A are integers. This implies that all the entries in the adjugate matrix of A are integers.

Step 3: Use Theorem 8

By Theorem 8,

\(\begin{array}{c}{A^{ - 1}} = \frac{1}{{\det A}}{\rm{adj}}\,A\\ = \frac{1}{1}{\rm{adj}}\,A\\{A^{ - 1}} = {\rm{adj}}\,A\end{array}\)

Hence, we conclude that all entries in \({A^{ - 1}}\) are integers.

Most popular questions for Math Textbooks

Compute the determinants of the elementary matrices given in Exercise 25-30.

29. \(\left[ {\begin{aligned}{*{20}{c}}1&0&0\\0&k&0\\0&0&1\end{aligned}} \right]\).

Is it true that \(det \left( {A + B} \right) = det A + det B\)? Experiment with four pairs of random matrices as in Exercise 44, and make a conjecture.

Find the determinant in Exercise 17, where \[\left| {\begin{aligned}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right| = {\bf{7}}\].

17. \[\left| {\begin{aligned}{*{20}{c}}{{\bf{a}} + {\bf{d}}}&{{\bf{b}} + {\bf{e}}}&{{\bf{c}} + {\bf{f}}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right|\]

Find the determinant in Exercise 18, where \(\left| {\begin{aligned}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right| = {\bf{7}}\).

18. \(\left| {\begin{aligned}{*{20}{c}}{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right|\)

Question: 13. Show that if A is invertible, then adj A is invertible, and \({\left( {adj\,A} \right)^{ - {\bf{1}}}} = \frac{{\bf{1}}}{{detA}}A\).

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