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Chapter 5: Q 5.3-25E (page 233)

Question: If A and B are arbitrary n 脳 n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

ATBTBA.

Short Answer

Expert verified

ATBTBAis symmetric.

Step by step solution

01

Condition to be a symmetric.

A matrix is symmetric if and only if it is equal to its transpose.

All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. A matrix is skew-symmetric if and only if it is the opposite of its transpose.

For example,

11-1120-105

Thus,

LetC=ATBT(AT)T(BT)T

Then,

C=ATBT(AT)T(BT)T鈬�CT=ATBTBAC=CT

Therefore,If A and B are arbitraryn 脳 n matrices then the matrices ATBTBAis symmetric.

Hence, the matrices is symmetric.

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Most popular questions from this chapter

Complete the proof of Theorem 5.1.4: Orthogonal projection is linear transformation.

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

11.[2306],[44213]

If A and B are arbitraryn脳n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

B(A+A螕)B螕.

If then脳n matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?-B.

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

4路[403],[250-25],[0-20]
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