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Chapter 5: Q 5.3-66E (page 235)

Question: Find an orthogonal 2脳2 matrix Asuch that all the entries of 100Aare integers while all the entries of 10A fail to be integers.

Short Answer

Expert verified

The matrix is A=B2=0.28-0.960.960.28.

Step by step solution

01

Orthogonal matrix.

Inlinear algebra, an orthogonal matrix, is a realsquare matrixwhose columns and rows areorthonormalvectors.

One way to express this is

QTQ=QQT=I

Where QTis thetransposeof Qand I

is theidentity matrix.

For example,

1001(identitytransformation)

02

Product of an orthogonal matrix.

a.The product ABof two orthogonal n脳nmatrices Aand Bis orthogonal.

b.The inverseA-1 of an orthogonal n脳nmatrix Ais orthogonal.

Now, take one of the solution from the previous exercise, say

B=0.8-0.60.60.8and then let

A=B2=0.28-0.960.960.28

Thus, the matrix A is orthogonal by the above theorem product of orthogonal matrices.

Hence, the matrix A is orthogonal and A=B2=0.28-0.960.960.28.

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Most popular questions from this chapter

Let Abe ann脳mmatrix. Is the formula(kerA)鈯�=im(AT)necessarily true? Explain.

Question: If the n脳nmatrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?AB2A.

Question: If the n脳nmatrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?role="math" localid="1659492178067" B-1.

Consider the orthonormal vectors u鈫�1,u鈫�2,u鈫�3,u鈫�4,u鈫�5 in鈩�10. Find the length of the vectorx鈫�=7u鈫�1-3u鈫�2+2u鈫�3+u鈫�4-u鈫�5.

Leonardo da Vinci and the resolution of forces. Leonardo (1452鈥�1519) asked himself how the weight of a body, supported by two strings of different length, is apportioned between the two strings.

Three forces are acting at the point D: the tensions F1and F2in the strings and the weight W鈫�. Leonardo believed that

||F鈫�1||||F鈫�2||=E炉A炉E炉B炉

Was he right? (Source: Les Manuscripts de L茅onard de Vinci, published by Ravaisson-Mollien, Paris, 1890.)

Hint: ResolveF1into a horizontal and a vertical component; do the same for F2. Since the system is at rest, the equationF鈫�1+F鈫�2+W鈫�=0鈫�holds. Express the ratios

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