Condition number equals its inverse. How could you show that $\kappa(A)$ = $\kappa(A^{-1})$?
And that for any nonzero scalar c, $\kappa(cA)$ = $\kappa(A)$?
Maybe a little explanation as to some intuition behind this as well.
 A: Consider the linear system
$$
Ax=b.
$$
From this equation, it follows immediately that (for any norm $\left\Vert \cdot\right\Vert $)
$$
\Vert b\Vert\leq\Vert A\Vert\Vert x\Vert.
$$
The condition number is a measure of how much the solution $x$ (assuming
$A$ is nonsingular) changes for a perturbation in the right-hand
side. Perturbing the right-hand side by $\Delta b$ gives
$$
A(x+\Delta x)=b+\Delta b.
$$
It follows that
$$
\Delta x=A^{-1}\Delta b.
$$
It follows that
$$
\Vert\Delta x\Vert\leq\Vert A^{-1}\Vert\Vert\Delta b\Vert.
$$
Therefore,
$$
\frac{\Vert\Delta x\Vert}{\Vert x\Vert}\leq\underbrace{\Vert A\Vert\Vert A^{-1}\Vert}_{\kappa(A)}\frac{\Vert\Delta b\Vert}{\Vert b\Vert}.
$$
This inequality says that the change in the solution (i.e. $\Delta x/x$)
is bounded above by some quantity that is $\kappa(A)$ times the change in the right-hand side (i.e. $\Delta b/b$). The definition above is symmetric in $A$ and $A^{-1}$, and the answer to your question follows.
More importantly, the above suggests the following:

If $\kappa(A)$ is large, a small change in the right-hand side might result in a large change in the solution.

You may have seen the condition number defined in terms of the singular
values of the matrix, which corresponds to taking the norm to be $\left\Vert \cdot\right\Vert _{2}$
(i.e. the Euclidean norm) in the above.
