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Chapter 6: Q2E (page 308)

Ifdet(A10)=(detA)10for all 10脳10matricesA.

Short Answer

Expert verified

Therefore, the given equation satisfies the condition by using Cauchy-Binet formula.

detA10=(detA)10.

Step by step solution

01

Matrix Definition

A matrixis a set of numbers arranged in rowsand columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are rows and columns, the matrix is said to be a 鈥� mby n鈥� matrix, written 鈥�m脳n .鈥�

02

Given 

Given equations,

detA10=(detA)10

03

To check whether the given condition is true or false

Given equations,

detA10=(detA)10

By using Cauchy鈭払inet formula,

A=Lf&B=Rg,for all f,g:m鈫�n.

Since the determinant of a permutation matrix equals the signature of the permutation, the identity follows from the fact that signatures are multiplicative.

Therefore, the given equation satisfies the condition by using Cauchy-Binet formula.

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Most popular questions from this chapter

det(4A)=4detAfor all4脳4 matricesA.

a. Find a noninvertible 2脳2matrix whose entries are four distinct prime numbers, or explain why no such matrix exists.

b. Find a noninvertible 3脳3matrix whose entries are nine distinct prime numbers, or explain why no such matrix exists.

Find the determinants of the linear transformations in Exercises 17 through 28.

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If all the entries of a square matrix are 1 or 0, thenmust be 1, 0, or -1.

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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