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Chapter 5: Q13E (page 233)

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? 3A.

Short Answer

Expert verified

The Matrix 3A is symmetric.

Step by step solution

01

Definition of symmetric matrix.

A square matrix is symmetric matrix ifA=A-T

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,kA=kAT it gives,

3AT=3AT=3A

Therefore, 3A is a symmetric matrix.

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Most popular questions from this chapter

Among all the vectors in Rnwhose components add up to 1, find the vector of minimal length. In the casen=2 , explain your solution geometrically.

Find the length of each of the vectors In exercises 1 through 3.

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Consider the orthonormal vectors u鈫�1,u鈫�2,u鈫�3,u鈫�4,u鈫�5 in鈩�10. Find the length of the vectorx鈫�=7u鈫�1-3u鈫�2+2u鈫�3+u鈫�4-u鈫�5.

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a.Find all n脳nmatrices that are both orthogonal and upper triangular, with positive diagonal entries.

b.Show that the QRfactorization of an invertible n脳nmatrix is unique. Hint: If,A=Q1R1=Q2R2 thenthe matrixA=Q2-1Q1=Q2R1-1 is both orthogonal and upper triangular, with positive diagonal entries.
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