Answer to Question #190371 in Linear Algebra for Regomoditswe Dibob

Question

Show that if A is a matrix with a row of zeros (or a column of zeros), then A cannot have an inverse 

Answer 1

Suppose the ith-column of 

A is zero. Then fix a family of vectors v(t) ∈ $\R^n$ , where

t ∈ $\R$  is the ith-entry of the vector v(t)

v(t) and all other entries are equal (e.g. all 1

1). Then since the ith-column of

A is zero all this vectors get mapped to the same vector under A.

So A is not injective, in particular not invertible (not bijective).