Singular / nonsingular question

  • Thread starter loli12
  • Start date
In summary, the conversation discusses how to prove that if matrix B is singular, then the product of matrices A and B (C = AB) must also be singular. The discussion includes a hint to use the theorem of equivalent conditions for nonsingularity and addresses a potential misunderstanding of the definition of singular. It is concluded that if B is singular, then there must exist a nonzero x such that Bx = 0, which also implies that ABx = 0 and therefore, C = AB is singular.
  • #1
loli12
Please help me on this...!

Let A and B be n x n matrices and let C = AB. Prove that if B is singular then C must be singular.
(The hint they provided is to use this theorem : Equivalent conditions for nonsingularity. 1. A is singular. 2. A x=0 has only the trivial soultion 0. 3. A is row equivalent to I )

Please please help!
 
Physics news on Phys.org
  • #2
So, if B is singular there is a nonzero x such that Bx=0, and, since for all matrices M0=0, then what is Cx?
 
  • #3
Oh, I understand now. Thanks~
 
  • #4
did you try that problem befiore asking for help?
 
  • #5
of coz i did, but did it in a wrong direction by dealing with the inverses.
 
  • #6
well, that also works: if C has an inverse then TC=1, say, so that TAB=1, hence B ins invertible, with inverse TA, contradiction, so C cannot posses an inverse.
 
  • #7
here is why i asked what i did.

you wrote:
C = AB. Prove that if B is singular then C must be singular.
(The hint they provided is to use this theorem : Equivalent conditions for nonsingularity. 1. A is singular. 2. A x=0 has only the trivial solution 0."


AHA! now I see your problem, you have the definitions backwards! this is the definition of non singular, rather than the definition of singualr. you could not possibly do the problem with this incorrect version of the notion of singular. i.e. 1 is not what it should be. was that the rpoblem?

if notm and you really understood that non singular meant that Ax = 0 impleis x =0, then,

1) did you understand that then B singualr means that there is some x which is not zero but with Bx = 0?

If so, then it is almost trivial to see that also (AB)x= A(Bx) = A0 = 0 , hence AB is singular.

I am having difficulty thinking you did not see how go from Bx = 0 to ABx = 0, so I was thinking the problem was elsewhere. Maybe in stating the definition of singular?

If I am wrong, then notice that if you see that Bx = 0,ma nd you are asking yourself if (AB)y everye quals zero for a non zero y, that x is a natural choice. Indeed it is the only choice you have from what is given. As in zenk, you must use whatever you are given.

The reason I ask is it is always helpful to see where you went astray, so as to observe how to avoid it next time.
 
  • #8
matt grime said:
well, that also works: if C has an inverse then TC=1, say, so that TAB=1, hence B ins invertible, with inverse TA, contradiction, so C cannot posses an inverse.

Do you mean with TC
[tex] C^T,[/tex]
with TAB
[tex] A^{T}B[/tex]
and with TA
[tex] A^T[/tex]?
 

1. What does "singular" and "nonsingular" mean in the context of a scientific question?

"Singular" refers to something that is unique or one-of-a-kind, while "nonsingular" refers to something that is not unique and has multiple instances or variations.

2. How are singular and nonsingular questions different?

Singular questions typically have a definitive answer, while nonsingular questions may have multiple answers or may be open-ended.

3. Can you give an example of a singular and a nonsingular question?

A singular question could be "What is the boiling point of water?" as there is only one correct answer. A nonsingular question could be "How does temperature affect the boiling point of different liquids?" as there could be multiple factors and variables that influence the answer.

4. Are there any advantages to asking a singular or nonsingular question in a scientific context?

Both types of questions have their advantages. Singular questions allow for more focused research and a clear-cut answer, while nonsingular questions can lead to a deeper understanding of a topic and more exploration of different ideas and perspectives.

5. Can a nonsingular question ever lead to a singular answer?

Yes, a nonsingular question can lead to a singular answer if the research and experimentation are thorough enough to eliminate all other possibilities and arrive at a definitive conclusion.

Similar threads

  • Special and General Relativity
Replies
7
Views
1K
Replies
9
Views
947
  • Linear and Abstract Algebra
Replies
7
Views
1K
  • Special and General Relativity
Replies
20
Views
691
Replies
2
Views
937
  • Linear and Abstract Algebra
Replies
2
Views
917
  • Linear and Abstract Algebra
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
937
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
995
Back
Top