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Chapter 3: Q18E (page 165)

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.

Short Answer

Expert verified

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

Step by step solution

01

Describe the given statement

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

02

Use the formula for cofactors

The cofactor ofA 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 ofA are integers.

03

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.

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Most popular questions from this chapter

Each equation in Exercises 1-4 illustrates a property of determinants. State the property

\(\left| {\begin{array}{*{20}{c}}{\bf{0}}&{\bf{5}}&{ - {\bf{2}}}\\{\bf{1}}&{ - {\bf{3}}}&{\bf{6}}\\{\bf{4}}&{ - {\bf{1}}}&{\bf{8}}\end{array}} \right| = - \left| {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{3}}}&{\bf{6}}\\{\bf{0}}&{\bf{5}}&{ - {\bf{2}}}\\{\bf{4}}&{ - {\bf{1}}}&{\bf{8}}\end{array}} \right|\)

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: In Exercise 16, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

16. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{2}}&{\bf{4}}\\{\bf{0}}&{ - {\bf{3}}}&{\bf{1}}\\{\bf{0}}&{\bf{0}}&{ - {\bf{2}}}\end{array}} \right)\)

In Exercise 19-24, explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.

\[\left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{1}}\\{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{\bf{1}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}k&{\bf{0}}&k\\{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{\bf{1}}\end{array}} \right]\]

Question: 12. Use the concept of area of a parallelogram to write a statement about a \(2 \times 2\) matrix A that is true if and only if A is invertible.
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