Inverse of a matrix + determinant

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Discussion Overview

The discussion revolves around the conditions for the invertibility of a matrix, specifically focusing on the relationship between a matrix's determinant and its invertibility. Participants explore theoretical aspects of matrix algebra, particularly in relation to determinants and inverse matrices.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that a matrix is invertible if its determinant is non-zero, while a zero determinant indicates that the matrix is not invertible.
  • One participant suggests that the term "matrix" in the context of the determinant being zero may have been a typo for "inverse matrix".
  • Another participant explains that the inverse of an invertible matrix can be expressed in terms of the minors of the original matrix and emphasizes that division by zero is problematic.
  • A further contribution outlines a property of determinants, stating that for the product of two matrices, the determinant of the product equals the product of their determinants, reinforcing that a zero determinant would prevent the existence of an inverse.

Areas of Agreement / Disagreement

Participants generally agree on the relationship between a matrix's determinant and its invertibility, but there are nuances in the language and definitions used, leading to some clarification and correction of terms.

Contextual Notes

There is a potential ambiguity in the use of the term "infinite" in relation to determinants, and the discussion does not resolve the implications of this terminology. Additionally, the discussion does not fully address the mathematical steps involved in determining invertibility beyond the determinant condition.

JamesGoh
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To determine if a matrix is invertible or not, can we determine this by seeing if the determinant of the matrix is zero or non-zero ?

If it's zero, then the matrix doesn't exist because the inverse of the determinant would be an infinite number ?
 
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JamesGoh said:
To determine if a matrix is invertible or not, can we determine this by seeing if the determinant of the matrix is zero or non-zero ?

Yes.

If it's zero, then the matrix doesn't exist because the inverse of the determinant would be an infinite number ?

I think "matrix" was a typo for "inverse matrix".

You need to be careful about how you use the word "infinite" in mathematics, but your basic idea is right.
 
A reason for that is that the inverse of an invertible matrix is the matrix in which the "ij" term is the minor of the "ji" term of the original matrix divided by the determinant. Given an "ji" term, it has a minor so the only problem is that we cannot divide by 0.
 
Another way of looking at it (for nxn-matrices):

Det(AB)=DetADetB;

In our case, say we have an inverse A' for A, then:

AA'=I , so that,

Det(AA')=DetADetA'=DetI=1,

So you need two numbers whose product equals one, and that rules out DetA=0.
 

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