Prove $k(AB) \leq k(A)k(B)$ where $k(\cdot)$ denotes the condition number
Solution 1:
You proof is correct. Both $A, B$ should be invertible, so $\|(AB)^{-1}\|=\|B^{-1}A^{-1}\|\le \|B^{-1}\|\|A^{-1}\| $.
Prove $k(AB) \leq k(A)k(B)$ where $k(\cdot)$ denotes the condition number
Solution 1:
You proof is correct. Both $A, B$ should be invertible, so $\|(AB)^{-1}\|=\|B^{-1}A^{-1}\|\le \|B^{-1}\|\|A^{-1}\| $.