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Chapter 6: Q38E (page 309)

If all the entries of matrices AandA-1 are integers, then the equationA=det(A-1) must hold.

Short Answer

Expert verified

Therefore,detA=detA-1 So, the given statement is true.

Step by step solution

01

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 鈥� m by n 鈥� matrix, written 鈥� m脳n.鈥�

02

To check whether the given condition is true or false

If A is ann脳nmatrix such that both A andA-1have integer entries, then both have integer determinants.

Then, from 1=detln=detAdetA-1.

We conclude that detA and detA-1can be either -1 or 1 .

Furthermore, this means we can have either detA=detA-1=-1 or detA=detA-1=1.

Either way,

detA=detA-1

So, the given statement is true.

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Most popular questions from this chapter

Explain why any patternPin a matrixA, other than the diagonal pattern, contains at least one entry below the diagonal and at least one entry above the diagonal.

For an invertiblenxnmatrix A, what is the relationship betweenadj(A)andadj(A-1) a?

In Exercises 5 through 40, find the matrix of the given linear transformation with respect to the given basis. If no basis is specified, use standard basis:for,

forandfor,.For the spaceof upper triangularmatrices, use the basis

Unless another basis is given. In each case, determine whetheris an isomorphism. Ifisn鈥檛 an isomorphism, find bases of the kernel and image ofand thus determine the rank of.

21. from to with respect to the basis.

Consider two positive numbers a and b. Solve the following system:

|ax-by=1bx+ay=0|.

What are the signs of the solutions x and y? How does x change as bincreases?

Vandermonde determinants (introduced by Alexandre-Th茅ophile Vandermonde). Consider distinct real numbers a0,a1,.....,an.. We define(n+1)脳(n+1) the matrix

A=[11....1a0a1....ana02a12....a12a0na1n....ann]

Vandermonde showed that

det(A)=鈭�i>j(ai-aj)

the product of all differences(ai-aj), where exceeds j.
a. Verify this formula in the case ofn=1.
b. Suppose the Vandermonde formula holds forn=1. You are asked to demonstrate it for n. Consider the function

f(t)=det[11...11a0a1...an-1ta02a12...an-1t2鈰�鈰�...鈰�鈰�a0na1n...an-1ntn]

Explain why f(t) is a polynomial of nthdegree. Find the coefficient k oftn using Vandermonde's formula fora0,...,an-1. Explain why

role="math" localid="1659522435181" f(a0)=f(a1)=...=f(an-1)=0

Conclude that

f(t)=k(t-a0)(t-a1)...(t-an-1)

for the scalar k you found above. Substitutet=an to demonstrate Vandermonde's formula.
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