# norm and invertibility

I'm currently solving some problems on matrix norms and one of these is asking me to show if a matrix is invertible or not. I wento trough the solution and at the end there was written : $\|A\|>0$ hence $A$ is invertible. The Norm is not specified. Maybe I didn't nderstand the answer well but is this true : $A$ invertible $\Leftrightarrow \|A\|>0$ ? If not, is there a relation between the norm beeing striclty positive and the invertibility of a Matrix?

• Any orthogonal projection $P$ has norm $\|P\| = 1 > 0$, but is invertible if and only if $P$ is just the identity. However, if you're interested in results relating invertibility to operator norms, $\|I-A\| < 1$ is a sufficient condition for invertibility, since the Neumann series $\sum_k (I-A)^k$ then converges absolutely to $A^{-1}$. – Branimir Ćaćić Aug 16 '13 at 15:08

There is no way for a norm to encode invertibility this way because $A$ non invertible equivalent to $||A||=0$ would imply $A=0$ because of the norms !
I think your solution is refering to the determinant and in this case $\det A\neq 0$ is equivalent to invertibility