lower bound for the condition number

Question

I have shown that if we have an invertible matrix $A \in \mathcal{M}_{N}(\mathbb{R})$ and $C \in \mathcal{M}_{N}(\mathbb{R})$ such that $A+C$ is singular then $cond(A) \geq \frac{\mid \mid A \mid \mid }{\mid \mid C \mid \mid }.$

Now given this matrix $ \left( \begin{array}{ccc} 1 & -1 & 1 \\ -1 & \alpha & \alpha \\ 1 & \alpha & \alpha \end{array} \right)$ I want to use this result to find a lower bound $l_{\alpha}$ for $cond_{\infty} A$ such that $ \underset{\alpha \rightarrow 0} {\lim} l_{\alpha} = + \infty .$ Any hint?

Answer 1

Take $C=\begin{pmatrix}0&0&0\\0&-\alpha&-\alpha\\0&-\alpha&-\alpha\end{pmatrix}$. If $\alpha$ is small, then $cond_{\infty}(A)\geq l_{\alpha}=\dfrac{3}{2|\alpha|}$.