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Chapter 5: Q16E (page 233)

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?A+B.

Short Answer

Expert verified

The Matrix A + B is symmetric.

Step by step solution

01

Definition of symmetric matrix.

A square matrix is symmetric matrix ifA=AT

02

Verification whether the given matrix is symmetric.

Given that A and B are symmetric matrices and B is invertible.

Since A is symmetric, then A=AT.

Then by properties of transpose,A+BT=AT+BT=A+B it gives,

Therefore, A + B is a symmetric matrix.

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Most popular questions from this chapter

Consider the vector

v炉=[1234]in R4

Find a basis of the subspace of R4consisting of all vectors perpendicular to v鈫�.

Consider a QRfactorization

M=QRShow that R=QTM.

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]

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.

(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.
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