# Geometric explanation of why only full rank matrices are invertible.

Is it possible to geometry to explain why only full rank matrices have inverses?

I understand the geometric explanation of inverses from this post (to undo the transformation of a matrix). And it is also clear that full rank matrices in a space are vectors that all go different directions (independent from each other). But I can't seem to combine these two geometric explanations together to explain why only full rank matrices have inverses.

• A non-full-rank matrix has notrivial kernel, i.e., there exists some nonzero $v$ with $Av=0$. So you'd need $A^{-1}0=v$, which is absurd – Hagen von Eitzen Sep 30 '16 at 21:23
• Matrices are called invertible, not inversable (this is hard to write, my computer immediately wants to convert this back to invertible). – Dietrich Burde Sep 30 '16 at 21:33
• @HagenvonEitzen Your explanation makes sense! – susanz Oct 1 '16 at 13:35