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Chapter 6: Q21E (page 309)

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

Short Answer

Expert verified

Therefore, the given condition is true.

Step by step solution

01

Orthogonal Matrix Definition. 

A square matrix with real numbers or elements is said to be an orthogonal matrix, if its transpose is equal to its inverse matrix.

Or we can say, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

02

To check whether the given condition is true or false.

For, 2脳2matrices whose entries are 0 and 1 , the determinant can only be -1, 0 , or 1.

Forn脳n matrices, wheren>2, this applies using the Laplace expansion.

Therefore,

detA=-1,0,1.

Therefore, the given condition satisfied and the given statement is true.

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Most popular questions from this chapter

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

26. T(M)=[1223]M+M[1223]from the space V of symmetric 2 脳 2 matrices to V

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.

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.

Consider a4脳4matrix A with rowsv鈫�1,v鈫�2,v鈫�3,v鈫�4. If det(A) = 8, find the determinants in Exercises 11 through16.

13. det[v鈫�1v鈫�2v鈫�3v鈫�4]

Show that an nxn matrixAhas at least one nonzero minor if (and only if) rank(A)鈮�n-1.
See all solutions

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