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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q53E (page 292)

Consider an invertible $2 脳 2$ matrix $A$ with integer entries.
a. Show that if the entries of $A^{- 1}$ are integers, thenlocalid="1660721234455" $d e t A = 1$ or $d e t A = - 1$ .
b. Show the converse: If $d e t A = 1$ or $d e t A = - 1$ , then the entries of $A^{- 1}$ are integers.

Short Answer

Expert verified

a) The given result is proved.

b) The $A^{- 1}$ is given by $A^{- 1} = 卤 [\begin{matrix} d & - b \\ - c & d \end{matrix}]$ ,

Step by step solution

01

Step by Step Solution: Step 1: Matrix Definition

Matrixis aset of numbers arranged inrowsandcolumnsso 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 a 鈥� $m$ by $n$ 鈥� matrix, written 鈥� $m 脳 n$ .鈥�

02

(a)Step 2: To showdetA=±1

If all entries of both A and $a^{- 1}$ are integers, then both determinants are integers, too. We have.

$1 = \det I 1 = \det (A A^{- 1}) 1 = \det A \det A^{- 1} .$

Then, $\det A^{- 1}$ can only be $- 1$ or $1$ .

03

(b)Step 3: To find A-1

We have,

$A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ ,with all integer entries.

If $\det A = 卤 1$ , then

$A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}] A^{- 1} = 卤 [\begin{matrix} d & - b \\ - c & d \end{matrix}]$

This again has all integer entries.

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Most popular questions from this chapter

Use Cramer's rule to solve the systems in Exercises 22 through 24.

22. $| \begin{matrix} 3 x + & 7 y = & 1 \\ 4 x + & 11 y = & 3 \end{matrix} |$

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

Consider the quadrilateral in the accompanying figure, with vertices $P_{i} = (x_{i}, y_{i})$ , for $i = 1, 2, 3, 4$ . Show that the area of this quadrilateral is

$\frac{1}{2} (d e t [\begin{matrix} x_{1} & x_{2} \\ y_{1} & y_{2} \end{matrix}] + d e t [\begin{matrix} x_{2} & x_{3} \\ y_{2} & y_{3} \end{matrix}] + d e t [\begin{matrix} x_{3} & x_{4} \\ y_{3} & y_{4} \end{matrix}] + d e t [\begin{matrix} x_{4} & x_{1} \\ y_{4} & y_{1} \end{matrix}])$ .

For an invertible nxn matrix A, what is the relationship between det (A) and det (A)?

IfA is a $4 脳 4$ matrix whose entries are all 1 or -1 , then $d e t A$ must be divisible by 8 (i.e., $d e t A = 8 k$ for some integer k).

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