Rank of a Matrix: Can We Eliminate Lines to Get Non-Vanishing Determinant?

[quasar987]

If A is an nxk matrix of real numbers (n>=k) of rank k, is it true that we can eliminate n-k lines of A to obtain a matrix A' of nonvanishing determinant?

I convinced myself of that one time while in the bus and now I can't find the proof.

 

[Defennder]

Hmm, if A is of rank k, then that means that the row space of A is spanned by k vectors , and this means that we can eliminate (n-k) rows of A which are effectively linear combinations of the others. So when we do that we have A', which is a kxk matrix and of rank k, which implies that it is invertible which in turn implies its det is non-zero.

 

[quasar987]

Ohhh.. yeah!

Thanks!

 

[Defennder]

Welcome. I'm trying to jog my linear algebra memory for a intermediate linear algebra class this semester.