Proving Non Singularity of Square Matrix is Necessary for Invertibility

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Discussion Overview

The discussion revolves around the proof that a square matrix A is invertible if and only if it is non-singular. Participants are exploring the definitions and implications of invertibility and non-singularity in the context of linear algebra, focusing on the necessary conditions and proofs related to these concepts.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • Some participants assert that a square matrix A is invertible if and only if it is non-singular, referencing the relationship between the determinant and the existence of an inverse.
  • One participant notes that if the determinant |A|=0, then the inverse A^ is not defined, suggesting that non-singularity is a necessary condition for invertibility.
  • Another participant emphasizes the need to use the definition of invertibility, which involves transforming a matrix into the identity matrix through elementary operations, as part of the proof.
  • Some participants highlight that the terms "invertible" and "non-singular" are often used interchangeably, which can lead to confusion in proofs.
  • There is a question regarding the completeness of the initial proofs provided, with participants seeking clarification on what might be missing to achieve a full proof.

Areas of Agreement / Disagreement

Participants generally agree that there is a connection between invertibility and non-singularity, but there is disagreement on the adequacy of the proofs presented and the definitions used. The discussion remains unresolved regarding the completeness and correctness of the initial answers.

Contextual Notes

Some limitations noted include the reliance on definitions that may not have been explicitly stated in the initial proofs, as well as the potential ambiguity in the use of terms like "invertible" and "non-singular." There is also mention of the need for a more rigorous approach to constructing the proof.

johncena
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Q:Prove that a square matrix A is invertible iff A is non singular.
My Ans: Since the inverse of a square matrix is given by,
A^ = (1/|A|)adj.A (Where A^ is A inverse)
If |A|=0, A^ is not defined.
i.e, A^ exist only if A is non singular. In other words, a square matrix A is invertible iff A is non singular.
Conversly, Let |A|=0, i.e., let A be singular.
then, A^=(1/|A|)adj.A = (1/0)adj.A (not defined)
Hence we conclude that A^ exist only if A is non singular.

I got only 1 out of 5 marks for this answer. What is missing in my answer?
 
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johncena said:
Q:Prove that a square matrix A is invertible iff A is non singular.
My Ans: Since the inverse of a square matrix is given by,
A^ = (1/|A|)adj.A (Where A^ is A inverse)
If |A|=0, A^ is not defined.
i.e, A^ exist only if A is non singular. In other words, a square matrix A is invertible iff A is non singular.
Conversly, Let |A|=0, i.e., let A be singular.
then, A^=(1/|A|)adj.A = (1/0)adj.A (not defined)
Hence we conclude that A^ exist only if A is non singular.

I got only 1 out of 5 marks for this answer. What is missing in my answer?

Let A be a square invertible matrix.
Then there exist a finite number of elementary operations on A that will transform A to I. That is, E1E2...EnA=I. Hence, there exists an inverse of A, namely E1E2...En. So A is nonsingular.
Thus, if A is invertible, then A is nonsingular.

Conversely, let A be nonsingular.
Then there exists an inverse of A.
Hence there exist a finite number of elementary operations on A such that A is transformed to I. That is A^-1 = E1E2...En and (A^-1)A=I. So A is invertible.
Thus, if A is nonsingular, then A is invertible.

Therefore, A is invertible iff A is nonsingular.
 
Why my answer is incorrect?
 
johncena said:
Why my answer is incorrect?

Strictly speaking, "invertible" means that a matrix can be transformed into the identity matrix by a finite series of elementary operations. Your proof did not use the definition.

One problem with constructing the proof is that invertible and nonsigular are often used interchangeably. But this is precisely because of the bi-implication.

Are you taking linear algebra this summer or are you rehashing an old exam?
 

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