Proof or reject condition number if square matrix

Question

I want to proof(if true) or reject: For all A Square matrix nxn K(10A) = 10K(A), i think it's wrong since: K(cA)=K(A) (refe Show property of condition number)

if K(cA) == K(A) which is != from 10K(A), is my proof is correct?

Answer 1

Your proof is correct. To go into a bit more detail, $$ \kappa(10A)=||10A|| \cdot ||(10A)^{-1}||=10||A|| \cdot 10^{-1}||A^{-1}||=||A|| \cdot ||A^{-1}|| = \kappa(A) \neq 10 \kappa(A). $$