Condition number is less than n

Question

Show that for an $n$ x $n$ orthogonal matrix $A$ that $\operatorname{Cond}(A) \leq n$.

I need to use: $$\|x\|_1 \leq \sqrt n$$

I know that $\operatorname{Cond}(A)=1$ for $A$ orthogonal matrix. Also given that: $\operatorname{Cond}(A)= \|A\|_1 \cdot \|A^{-1}\|_1$

Answer 1

Since $A$ is orthogonal, $A^{-1}= A^T$

given $|| x||_1≤\sqrt n$

$Cond(A)= ||A||_1 * ||A^{-1}||_1 = ||A||_1 *||A^T||_1 ≤ \sqrt n * \sqrt n = n$