Page 224 - Matrix Analysis & Applied Linear Algebra
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4.5 More about Rank 219
Exercises for section 4.5
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4.5.1. Verify that rank A A = rank (A)= rank AA for
1 3 1 −4
A = −1 −3 1 0 .
2 6 2 −8
4.5.2. Determine dim N (A) ∩ R (B) for
−211 1 3 1 −4
A = −422 and B = −1 −31 0 .
000 2 6 2 −8
4.5.3. For the matrices given in Exercise 4.5.2, use the procedure described
on p. 211 to determine a basis for N (A) ∩ R (B).
4.5.4. If A 1 A 2 ··· A k is a product of square matrices such that some A i is
singular, explain why the entire product must be singular.
m×n T
4.5.5. For A ∈ , explain why A A = 0 implies A = 0.
4.5.6. Find rank (A) and all nonsingular submatrices of maximal order in
2 −1 1
A = 4 −2 1 .
8 −4 1
4.5.7. Is it possible that rank (AB) < rank (A) and rank (AB) < rank (B)
for the same pair of matrices?
4.5.8. Is rank (AB)= rank (BA) when both products are defined? Why?
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4.5.9. Explain why rank (AB)= rank (A) − dim N B ∩ R A .
4.5.10. Explain why dim N (A m×n B n×p ) = dim N (B) + dim R (B) ∩ N (A).
