Singularity of a matix

Question

If $A$ is any $n\times n$ non-singular matrix, then $\mathrm{cond}(A) = \mathrm{cond}(A^{-1})$? True or False?

I'm not sure how to answer this question.

Answer 1

Generally, we define $\mathrm{cond}(A)$ as

$$\mathrm{cond}(A):=\vert\vert A\vert\vert\cdot \vert\vert A^{-1}\vert\vert.$$

So we do have

$$\mathrm{cond}(A)=\vert\vert A\vert\vert\cdot \vert\vert A^{-1}\vert\vert=\vert\vert A^{—1}\vert\vert\cdot \vert\vert A\vert\vert=\mathrm{cond}(A^{-1}).$$