Is a Matrix with a Zero Column Invertible?

[lolimcool]

Can a square matrix with a column of all zero's be invertible?

 

[SteamKing]

It's not looking like it will be. How did you form the matrix in the first place?

 

[lolimcool]

i didnt form a matrix, I am just wondering is it possible for a square matrix to have a column and be invertible

 

[SteamKing]

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.

 

[Mark44]

Can a square matrix with a column of all zero's be invertible?

It's not looking like it will be.

The answer is a definite no.

 

[Sankaku]

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.

 

[jbunniii]

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?)

 

[HallsofIvy]

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.