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.