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Chapter 5: Q33E (page 264)

TRUE OR FALSE

If Ais an invertible matrix such that A-1=A, then Amust be orthogonal.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Definition of an orthogonal matrix

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 it can be said that, 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

Consider an example

For example,

Suppose A is a square matrix with real elements and ofn脳norder andAT is the transpose of A. Then according to the definition, if,AT=A-1is satisfied, then,

A路AT=I

If A is an invertible matrix such that A-1=A, thenA2=I.

This means that A is an involuntary matrix.

Consider the matrix

A=3-24-3鈬�A2=1001

Because its column vectors are not unit vectors. Thus, it can鈥檛 be an orthogonal matrix.

Hence, the statement is false.

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Most popular questions from this chapter

If u鈬赌is a unit vector in Rnand L=span(u鈬赌), then projL(x鈬赌)=(x鈬赌.u鈬赌)u鈬赌.for all the vectorsx鈬赌inRn.

If Ais ann脳mmatrix, is the formulaim(A)=im(AAT)necessarily true? Explain.

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

19.[212115]

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

8.[5422],[367-2]

Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? [0.60.80.80.6].
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