What is $\left | \left | A \right | \right |$ equals to in linear algebra?

Question

Can someone please tell me what is this $\left | \left | A \right | \right |$ equals to? (determinant inside determinant)

Answer 1

$||A||$ in a sense of $\det(\det( A))$ makes absolutely no sense. Even if you define $\det(A) := (\det (A))$, i.e. as a $1\times 1$-matrix, $\det (\det A)$ would be just $\det (A)$...

The determinant is a scalar.

Answer 2

Usually, in linear algebra (and functional analysis, and many other fields) double vertical lines $\|\cdot\|$ denote a norm. If $A:V\to W$ is a linear operator between normed vector spaces (or a matrix in the finite-dimensional case with a fixed choice of basis), then often the following norm is used $$\|A\| = \sup_{x\in V}\frac{\|Av\|_W}{\|v\|_V},$$ where $\|\cdot\|_V$ is the norm on $V$, and similarly for $W$.