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Chapter 5: Q18E (page 263)

If matrices Aand Bcommute, then matrices ATand BTmust commute 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+BT For all m脳nmatrices Aand B.
  2. (kA)T=kAT For 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

By the above theorem, consider the terms below.

ATBT=(BA)T=(AB)T=BTAT

Hence, the statement is true.

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Most popular questions from this chapter

TRUE OR FALSE? The equation det(AT)=det(A)holds for all 2脳2matrices A.

Question: Consider an n脳mmatrix A, a vector in Rm, and a vector w鈫�in Rm. Show that (Av鈫�).w鈫�=v鈫�.(ATw鈫�).

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.

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

2. v鈬赌=[234].

Consider a basisv鈫�1,v鈫�2,...v鈫�m of a subspaceVofrole="math" localid="1659434380505" 鈩�n. Show that a vectorx鈫� inrole="math" localid="1659434402220" 鈩�nis orthogonal toV if and only ifx鈫� is orthogonal to all vectorsv鈫�1,v鈫�2,...v鈫�m.
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