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Chapter 5: Q17E (page 263)

If Aand Bare symmetric n脳nmatrices, then A B B Amust be symmetric as well.

Short Answer

Expert verified

True

Step by step solution

01

Step by step solution Step 1: Properties of the transpose

Consider the properties of the transpose below.

  1. (A+B)T=AT+BTFor all m脳 nmatrices Aand B.
  2. (kA)T=kATFor all m脳nmatrices Aand for all scalars k.
  3. (AB)T=BTAT For all m脳pmatrices Aand for all matrices B.
  4. rank(AT)=rank(A)For all matrices A.
  5. (AT)-1=(A-1)T For all invertible n脳nmatrices A
02

Determine whether the statement is true or false

Consider the terms below.

(ABBA)T=(BA)T(AB)T=ATBTBTAT=ABBA

Hence, the statement is true.

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Most popular questions from this chapter

All nonzero symmetric matrices are invertible.

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

(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.

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.

a.Consider a vectorv鈯� in Rn, and a scalar k. Show that||kv鈫�||=|k|.||v鈫�||

b.Show that ifv鈯� is a nonzero vector in Rn, then

u鈫�=1||v鈫�||is a unit vector.
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