Looking clues for this problem.
Find all the matrices such that $\kappa(A) = 1$
We define $\kappa(A) = \|A\|\,\|A^{-1}\|$. If I'm looking matrices such that $\kappa(A) = 1$, I was thinking in this:
1) If $A = I$, the identity matrix, then $\kappa(A) = 1$.
2) If A is orthogonal, then the columns of $A$ form an orthonormal basis (orthonormal if we consider the euclidian norm) of $\mathbb{R}^{n \times n}$, then $\kappa(A) = 1$.
I'm not sure how to consider the norms $\| \cdot \|_1,\| \cdot \|_{\infty}$, because case (2) is valid whne we consider the euclidian norm.
Thanks for all your help!