Is a Matrix with a Zero Column Invertible?

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Discussion Overview

The discussion centers on the question of whether a square matrix with a column of all zeros can be invertible. Participants explore theoretical implications and properties related to matrix invertibility, determinants, and linear independence.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that a square matrix with a column of all zeros cannot be invertible, suggesting that the determinant would indicate this.
  • One participant questions the formation of such a matrix, implying that the context of its creation may influence its properties.
  • Another participant highlights the relationship between invertibility and linear independence, noting that a column of zeros indicates linear dependence.
  • A further explanation involves the mapping of vectors, where a matrix with a zero column maps multiple vectors to the zero vector, suggesting non-invertibility.

Areas of Agreement / Disagreement

Participants generally agree that a square matrix with a column of all zeros is not invertible, though the discussion includes various approaches and reasoning to support this view.

Contextual Notes

Some arguments rely on the properties of determinants and linear independence, while others focus on specific vector mappings. The discussion does not resolve all nuances of matrix theory.

lolimcool
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Can a square matrix with a column of all zero's be invertible?
 
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It's not looking like it will be. How did you form the matrix in the first place?
 
i didnt form a matrix, I am just wondering is it possible for a square matrix to have a column and be invertible
 
If you try to calculate the determinant of a matrix with a zero column by using minors, you will quickly see that a regular inverse does not exist. The same situation will apply if you have a matrix containing a row with all zeroes.
 
lolimcool said:
Can a square matrix with a column of all zero's be invertible?

SteamKing said:
It's not looking like it will be.

The answer is a definite no.
 
lolimcool said:
Can a square matrix with a column of all zero's be invertible?

There are many ways to show it is not possible.

Do you know the link between invertibility and determinant? If so, think about the quickest way to calculate the determinant of a matrix with a column (or row) of zeros.
 
Determinant-free answer:

In order for a square matrix to be invertible, its columns have to be linearly independent, which is clearly not the case if one of the columns is all zeros. (Why?)
 
Yet another way of looking at it (though almost the same as jbunnii):
If A has its "ith" column all 0s, then Av where v is all 0s except for the "ith" position, is the 0 vector. A cannot be invertible because there are many different vectors (differing in the "ith" place) that are all mapped into 0.
 

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