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Chapter 6: Q68E (page 293)

Using the terminology introduced in the proof of Theorem6.2.10, show thatsgnP=(-1)i+jsgn(Pij) . See Exercise 67.

Short Answer

Expert verified

It is proved thatsgnP=(-1)i+jsgnPi,j

Step by step solution

01

Given

To show that,

SgnP=(-1)i+jSgnPi,j

02

To Prove

To prove,

SgnP=(-1)i+jSgnPi,j

For a pattern P and i,j1,...,n , using Ex. 67,

We compute:

role="math" localid="1660714146091" sgnP=-1inversioninP}|sgnP=-1inversioninPai,j}|(-1)inversioninPnotincludingai,j]|sgnP=-1i+jsgnPi,j

It is proved that sgnP=-1i+jsgnPi,j

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Most popular questions from this chapter

For every nonzero2脳2 matrix A there exists a 2脳2 matrix B such that det(A+B)鈮�detA+detB.

Consider a 4x4 matrix A with rows V1鈫�,V2鈫�,V3鈫�,V4鈫�. If det(A) = 8, find the determinants in Exercises 11 through 16.

16. role="math" localid="1659506283449" det[6v鈬赌1+2v鈬赌4v鈬赌2v鈬赌33v鈬赌1+v鈬赌4]

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

3.[13241648130026412]

There exists a 4脳4 matrix whose entries are all 1or -1 , and such thatdetA=16.

Question:We say that a linear transformation Tfrom N3toN3to preserves orientation if it transforms any positively oriented basis into another positively oriented basis. See Exercise 19. Explain why a linear transformationT(x鈫�)=Ax鈫�preserves orientation if (and only if) detAis positive.
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