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Chapter 5: Q32E (page 233)

(a) Consider an matrix A such that A螕A=Im. It is necessarily true that? Explain.

(b) Consider an n脳nmatrix A such that ATA=In. Is it necessarily true that AAT=In? Explain.

Short Answer

Expert verified
  1. Not always true for condition (a).
  2. Always true for condition (b).

Step by step solution

01

Determine AAT≠I3.

To obtain AAT鈮�I3consider the matrix below.

A=100100

In the matrix A. performing product,

AAT=I2.

Which shows that AAT鈮�I3

Thus, according to condition (a) it is not necessarily true that AAT=I3.

02

Consider the definition of inverse matrix.

The inverse of matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A-1.

And the formula of inverse matrix is given below.

A-1=1A.AdjA

For example,

A=1-234A=0.40.2-0.30.1

Thus, by consider the definition of inverse matrix above the condition (b) is always true.

Hence, condition (a) is not necessarily true and condition (b) is always true.

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Most popular questions from this chapter

To make a trend analysis of six evenly spaced data points, one can use orthogonal polynomials with respect to evaluation at the points \(t = - 5, - 3, - 1,\,\,1,\,\,3,{\rm{ and }}5\).

  1. Show that the first three orthogonal polynomials are

\({p_0}\left( t \right) = 1,\,\,\,\,\,\,{p_1}\left( t \right) = t,{\rm{ and }}{p_2}\left( t \right) = \frac{3}{8}{t^2} - \frac{{35}}{8}\)

(The polynomial \({p_2}\) has been scaled so that its values at the evaluation points are small integers.)

  1. Fit a quadratic trend function to the data \(\left( { - 5,1} \right),\left( { - 3,1} \right),\left( { - 1,4} \right),\left( {1,4} \right),\left( {3,6} \right),\left( {5,8} \right)\).

Consider a symmetric n脳mmatrix A. What is the relationship between Im(A)and ker(A)?

Consider the linear systemAx鈫�=b鈫� , where

A=[1326]and b鈫�=[1020].

a. Draw a sketch showing the following subsets of 鈩�2:

  • The kernel ofA , and(kerA)鈯�
  • The image of AT
  • The solution setSof the system Ax鈫�=b鈫�

b.What relationship do you observe between(kerA) and im(AT)? Explain.

c. What relationship do you observe betweenrole="math" localid="1660916844921" ker(A) and S? Explain.

d. Find the unique vectorx鈫�0 in the intersection ofS and(kerA)鈯� . Show x鈫�0on your sketch.

e. What can you say about the length of x鈫�0compared with the length of all other vectors in S?

Let S(t)be the number of daylight hours on the tth day of the year 2012 in Rome, Italy. We are given the following data for S(t):

We wish to fit a trigonometric function of the form

f(t)=a+bsin(2蟿蟿366t)+ccos(2蟿蟿366t)

To these data. Find the best approximation of this form, using least squares. How many daylight hours does your model predict for the longest day of the year 2012? (The actual value is 15 hours, 13 minutes, 39 seconds.)

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

15.[21-2]
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