If A is triangular and no entry on the main diagonal is zero

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Discussion Overview

The discussion centers on the invertibility of a triangular matrix A, specifically addressing the conditions under which A is invertible when no entry on the main diagonal is zero. The scope includes theoretical aspects of linear algebra and properties of determinants.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants assert that a matrix is invertible if and only if its determinant is not zero, referencing the property of triangular matrices where the determinant is the product of the diagonal entries.
  • There is a clarification that the determinant must be non-zero for invertibility, with one participant confirming this understanding.
  • Another participant mentions that the independence of rows and columns implies that the linear map defined by the matrix is injective.

Areas of Agreement / Disagreement

Participants generally agree on the condition that a triangular matrix is invertible if its determinant is not zero, but there is some repetition in the statements made, indicating a lack of consensus on the clarity of the initial explanation.

Contextual Notes

Some assumptions about the properties of triangular matrices and determinants are present, but not all participants may have explicitly stated these assumptions, leading to potential ambiguity in the discussion.

muthuraman
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A is invertible... How give me the proper explanation...
 
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A matrix is invertible if and only if its determinant is 0. And for any triangular matrix, the determinant is simply the product of the numbers on the diagonal.
 
HallsofIvy said:
A matrix is invertible if and only if its determinant is 0. And for any triangular matrix, the determinant is simply the product of the numbers on the diagonal.
I presume you meant determinant is NOT zero.
 
mathman said:
I presume you meant determinant is NOT zero.
Yes, of course. Thank you!
 
also it should be obvious that the rows are independent. also the columns. so the map defined by the matrix is clearly injective.
 

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