Prove both sufficiency and necessity

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In summary, the conversation discusses proving that the product of two n x n matrices, AB, is invertible if and only if both A and B are invertible. The sufficiency and necessity of this statement is also addressed. The conversation also delves into the composition of bijections and the relationship between surjective and injective functions in this context. Finally, there is mention of using dimension theory and the determinant approach to prove these concepts.
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I need some help with this prove

Given that A and B are n x n matrices, prove the following : the product AB is invertible if and only if both A and B are invertible. (Prove both sufficiency and necessity)
 
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  • #2
What can you say about Det(AB) if it is invertible?
 
  • #3
can you prove the composition of bijections is a bijection?

and semi conversely, if the composition is a bijection, then the last function is surjective and the first function is injective?

this does most of it. the rest needs some dimension theory, i.e. that a linear map from R^n to itself is injective iff surjective.

the determinant approach is more sophisticated, but very slick and quick.
 
  • #4
mathwonk said:
can you prove the composition of bijections is a bijection?

and semi conversely, if the composition is a bijection, then the last function is surjective and the first function is injective?

this does most of it. the rest needs some dimension theory, i.e. that a linear map from R^n to itself is injective iff surjective.

the determinant approach is more sophisticated, but very slick and quick.

Just curious, regarding the bijections, are you aiming at the fact that for every matrix there exists a linear operator such that this very matrix is the matrix (of course) representation of that operator?
 
  • #5
yes. multiplication by the matrix is that operator. nread my lin ear algebra book, free online, 15 pages.
 

1. What does it mean to prove both sufficiency and necessity?

Proving both sufficiency and necessity means showing that a particular condition is both necessary and sufficient for a desired outcome or result. In other words, the condition is required for the outcome to occur (necessity) and it is enough to guarantee the outcome (sufficiency).

2. Why is it important to prove both sufficiency and necessity in scientific research?

In scientific research, proving both sufficiency and necessity is important because it ensures that the results or conclusions are valid and reliable. It helps to eliminate any potential confounding variables and provides a clear understanding of the relationship between the condition and the outcome.

3. How do you prove sufficiency and necessity in an experiment?

A common method for proving sufficiency and necessity in an experiment is by using a control group and an experimental group. The control group does not receive the condition being tested, while the experimental group does. If the outcome is only observed in the experimental group, it is considered sufficient. If the outcome is not observed in the control group, it is considered necessary.

4. Can a condition be necessary but not sufficient?

Yes, a condition can be necessary but not sufficient. This means that the condition is required for the outcome to occur, but it is not enough on its own to guarantee the outcome. Other factors may also be necessary for the outcome to be achieved.

5. How does proving both sufficiency and necessity contribute to the overall understanding of a scientific concept?

Proving both sufficiency and necessity provides a comprehensive understanding of a scientific concept by demonstrating the cause-and-effect relationship between a condition and an outcome. It helps to support or disprove a hypothesis and can lead to further research and advancements in the field.

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