# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Wed, 19 Jun 2013 04:55:39 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 7 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Singularity proofsPosted: Wed, 26 Jan 2011 02:43:43 UTC

Joined: Wed, 9 Jan 2008 19:24:20 UTC
Posts: 8
Alright im having some issues here with proofs (i think). I have the following problem.

Code:
Let A and B be nxn matrices and let C = A-B. Show that if A(Xo) = B(Xo) and (Xo) !=0 then C must be singular

I have attempted to prove it and am unsure if i have come to the correct conclusion, or if im even mathematically allowed to do what i have done

C = A - B
A(Xo) = B(Xo)
A(Xo)-B(Xo) = 0

C(Xo) = (A-B)Xo
C(Xo) = 0

Since Xo cannot be zero than C must be singular. Does that make sense???

In addition to that problem I have another one that im not sure if the proof is correct. The problem states....

Code:
Let A and B be nxn matrices and let C=AB. Prove that is B is singular then C must also be singular.

Attempt at a proof....

C=AB
I=I
C(C^-1) = I
AB(C^-1) = I
C^-1 = I(AB)^-1
C^-1 = B^-1 * A^-1

Since we know that B is singular B^-1 is undetermined, meaning that C^-1 must also be undetermined.  And that proves that C must be singular????  I think?

I feel like neither answer is correct....

Top

 Post subject: Re: Singularity proofsPosted: Wed, 26 Jan 2011 06:45:07 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 1 Oct 2003 04:45:43 UTC
Posts: 9642
dirty l3um wrote:
Let A and B be nxn matrices and let C = A-B. Show that if A(Xo) = B(Xo) and (Xo) !=0 then C must be singular

C = A - B
A(Xo) = B(Xo)
A(Xo)-B(Xo) = 0

C(Xo) = (A-B)Xo
C(Xo) = 0

In order for C to be nonsingular, its kernel must be trivial (i.e., contain only the zero vector).

dirty l3um wrote:
Let A and B be nxn matrices and let C=AB. Prove that is B is singular then C must also be singular.

Because B is singular, there is a nonzero vector x in its kernel, so Bx = 0. Consequently Cx = ABx = 0, so x belongs to the kernel of C also.

Top

 Post subject: Posted: Thu, 27 Jan 2011 04:49:57 UTC

Joined: Wed, 9 Jan 2008 19:24:20 UTC
Posts: 8
so does that mean that if in the first case to prove it is singular i could do something like.....
(X1 being a non zero number...)

A(Xo) = B(Xo)
(A-B)(Xo) = 0

C = A - B
C(Xo) = X1
(A-B)(Xo) = X1

this is a contradiction???? If not why not? Im very confused at this point.

Top

 Post subject: Posted: Thu, 27 Jan 2011 04:59:57 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 1 Oct 2003 04:45:43 UTC
Posts: 9642
It's a contradiction because, for any nonsingular matrix C, we have C(Xo) nonzero if and only if Xo is nonzero.

Top

 Post subject: Posted: Thu, 27 Jan 2011 18:21:01 UTC

Joined: Wed, 9 Jan 2008 19:24:20 UTC
Posts: 8
Matt, i think i understand the basis of what you are saying, however im having a very difficult time transposing that into a mathematical proof.... How would i prove it? (with equations)

Top

 Post subject: Posted: Thu, 27 Jan 2011 19:31:12 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12169
Location: Austin, TX
dirty l3um wrote:
Matt, i think i understand the basis of what you are saying, however im having a very difficult time transposing that into a mathematical proof.... How would i prove it? (with equations)

What do you mean "[transcribing] that into a mathematical proof?" that is a proof. It's nothing but definitions, so it should be transparent.

By definition, if B is singular, there is such that . But then , so C is also singular, with a singular vector

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Posted: Fri, 28 Jan 2011 15:42:54 UTC

Joined: Wed, 9 Jan 2008 19:24:20 UTC
Posts: 8
Thankyou for your help shadow and matt. I believe the problem was that i wasnt entirely sure what a proof was

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 7 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous