For $n>2$, are there  norms $\parallel.\parallel_{a}$  and  $\parallel.\parallel_{b}$ on $M_{n}(\mathbb{R})$ with the following property:

>$A\in M_{n}(\mathbb{R})$ is singular if and only if $\parallel A \parallel_{a}=\parallel A \parallel_{b}$ 

For $n=2,\;$  these norms are  $\parallel A \parallel_{a}=\sqrt{\sum a_{ij}^{2}}$ and
$\parallel A \parallel_{b}=\parallel.\parallel_{op}$, the operator norm.