{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

3.2 Rank of a Matrix

# 3.2 Rank of a Matrix - 3.2 3.2 The Rank of a Matrix and...

This preview shows pages 1–4. Sign up to view the full content.

3.2. 3.2. The Rank of a Matrix and Matrix Inverse 1. (a) F (Theorem 3.5) (b) F ( ) If A M m × n ( F ) , B M n × n ( F ) and B is invertible, then the rank(AB)=rank(A) (Example) A = B = 0 1 0 0 and rank ( A ) = rank ( B ) = 1 AB = 0 0 0 0 and rank ( AB ) = 0 (c) T (d) T (p.153, Corollary to the Theorem 3.4) (e) F (p.153, Corollary to the Theorem 3.4) (f) T (p.153, Theorem 3.4 and Theorem 3.5) (g) T (p.161) (h) T ( ) A M m × n ( F ) , rank ( A ) = dimR ( L A ) n (i) T 2. (a) A = 1 1 0 0 1 1 0 0 0 , rank ( A )=2 162 PNU-MATH

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
3.2. (b) A = 1 1 0 0 - 1 1 0 0 1 , rank ( A )= 3 (c) A = 1 0 2 0 1 2 , rank ( A )= 2 (d) A = 1 2 1 0 0 0 , rank ( A )= 1 (e) A = 1 2 0 1 1 0 0 1 1 - 2 0 0 0 0 1 0 0 0 0 0 , rank ( A )= 3 (g) A = 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 , rank ( A )= 1 3. A M m × n ( F ) , rank ( A ) = 0 iff A is the zero matrix ( ) clear ( ) let A = ( e 1 , 0 , · · · , 0) , e 1 6 = 0 Since rank ( A ) = 0 , e 1 is dependent a F s.t e 1 = a 0 It’s contradict to e 1 6 = 0 4. (a) 1 1 1 2 2 0 - 1 2 1 1 1 2 1 0 - 1 2 1 0 1 3 2 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 , 2 163 PNU-MATH
3.2. (b) 2 1 - 1 2 2 1 2 0 0 1 0 0 12 0 0 1 0 0 , 2 5. ( A | I n ) ˆ ( I n | B ) , B = A - 1 (a) 1 2 1 0 1 1 0 1 1 0 -1 2 0 1 1 -1 A - 1 = - 1 2 1 - 1 , rank ( A ) = 2 (b) 1 2 1 0 2 4 0 1 1 2 1 0 0 0 -2 1 rank ( A ) = 1, so @ A - 1 (c) 1 2 1 1 0 0 1 3 4 0 1 0 2 3 -1 0 0 1 1 0 -5 3 -2 0 0 1 3 -1 1 0 0 0 0 -3 1 1 rank ( A ) = 2, so @ A - 1 (d) 0 -2 4 1 0 0 1 1 -1 0 1 0 2 4 -5 0 0 1 1 0 0 1/6 15/9 -1/3 0 1 0 -5/18 -4/9 2/9 0 0 1 1/9 -2/9 1/9 rank ( A ) = 3 , A - 1 = 1 6 15 9 - 1 3 - 5 18 - 4 9 2 9 1 9 - 2 9 1 9 (e) 0 -2 4 1 0 0 1 1 -1 0 1 0 2 4 5 0 0 1 1 0 0 1/6 -1/3 1/2 0 1 0 1/2 0 -1/2 0 0 1 1/6 1/3 1/2 rank ( A ) = 3 , A - 1 = 1 6 - 1 3 1 2 1 2 0 - 1 2 - 1 6 1 3 1 2 (f) 0 2 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 0 -1/2 -1/2 0 0 0 0 -1/2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}