Page 146 - Matrix Analysis & Applied Linear Algebra

140 Chapter 3 Matrix Algebra

3.9.2. Consider the two matrices
   
2 2 0 −1 2 −6 8 2
A =  3 −14 0  and B =  5 1 4 −1  .
0 −88 3 3 −912 3
(a) Are A and B equivalent?
(b) Are A and B row equivalent?
(c) Are A and B column equivalent?

row
3.9.3. If A ∼ B, explain why the basic columns in A occupy exactly the
same positions as the basic columns in B.

3.9.4. A product of elementary interchange matrices—i.e., elementary matrices
of Type I—is called a permutation matrix. If P is a permutation
T
matrix, explain why P −1 = P .

3.9.5. If A n×n is a nonsingular matrix, which (if any) of the following state-
ments are true?
col
row
(a) A ∼ A −1 . (b) A ∼ A −1 . (c) A ∼ A −1 .
row col
(d) A ∼ I. (e) A ∼ I. (f) A ∼ I.
3.9.6. Which (if any) of the following statements are true?
row
T
T
∼ B .
(a) A ∼ B =⇒ A ∼ B . (b) A ∼ B =⇒ A T row T
row T col T row
(c) A ∼ B =⇒ A ∼ B . (d) A ∼ B =⇒ A ∼ B.
col row
(e) A ∼ B =⇒ A ∼ B. (f) A ∼ B =⇒ A ∼ B.
3.9.7. Show that every elementary matrix of Type I can be written as a product
of elementary matrices of Types II and III. Hint: Recall Exercise 1.2.12
on p. 14.

3.9.8. If rank (A m×n )= r, show that there exist matrices B m×r and C r×n
such that A = BC, where rank (B)= rank (C)= r. Such a factor-
ization is called a full-rank factorization. Hint: Consider the basic
columns of A and the nonzero rows of E A .

3.9.9. Prove that rank (A m×n ) = 1 if and only if there are nonzero columns
u m×1 and v n×1 such that
T
A = uv .
2
3.9.10. Prove that if rank (A n×n )=1, then A = τA, where τ = trace (A).