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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

Question: In Exercises 1 and 2, you may assume that\(\left\{ {{{\bf{u}}_{\bf{1}}},...,{{\bf{u}}_{\bf{4}}}} \right\}\)is an orthogonal basis for\({\mathbb{R}^{\bf{4}}}\).

1.\({{\bf{u}}_{\bf{1}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{ - {\bf{4}}}\\{ - {\bf{1}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{2}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{5}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{3}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{1}}\\{ - {\bf{4}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{4}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{5}}\\{ - {\bf{3}}}\\{ - {\bf{1}}}\\{\bf{1}}\end{aligned}} \right]\),\({\bf{x}} = \left[ {\begin{aligned}{*{20}{c}}{{\bf{10}}}\\{ - {\bf{8}}}\\{\bf{2}}\\{\bf{0}}\end{aligned}} \right]\)

Write x as the sum of two vectors, one in\({\bf{Span}}\left\{ {{{\bf{u}}_1},{{\bf{u}}_2},{{\bf{u}}_3}} \right\}\)and the other in\({\bf{Span}}\left\{ {{{\bf{u}}_{\bf{4}}}} \right\}\).

The formulaA(ATA)-1 for the matrix of an orthogonalprojection is derived in Exercise 67. Now considerthe QRfactorization of A, and express the matrixA(ATA)-1ATin terms of Q.

Consider an n x m matrix A with rank (A) = m. Is it always possible to write A as A = QL where Q is an n x m matrix with orthonormal columns and L is a lower triangular m x m matrix with positive diagonal entries? Explain.

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

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.)
See all solutions

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