Is determinant to a matrix analogous to the absolute value of a number?
We like to compare matrix properties to numbers, such as complex numbers. E.g. Hermitian matrix to all matrices is analogous as real numbers to complex numbers. I noticed some properties that determinants share with absolute values, such as $$|A+B|\le|A|+|B| \\ |AB|=|A|\cdot|B|$$ Can we somehow "prove" (or to put it better, provide some insight of) that determinants can be compared to absolute values?
Another question is, how we treat trace. What operation on numbers, or geometric objects can be compared to trace of a matrix? Or in other words, how do we understant trace more simply? I think the most important properties about trace are $$\mathrm{tr}(AB)=\mathrm{tr}(BA) \\ \mathrm{tr}(A+B) = \mathrm{tr}(A) + \mathrm{tr}(B)$$
The determinant of a square matrix is a multilinear function. The trace is a linear function. Trace also has a generalization to operators on C$^*$-algebras. A complex matrix algebra is an example of a C$^*$-algebra. We say that $A \le B$ if $B-A= C^*C$ for some matrix C. The absolute value of a square matrix $A$ is defined by $|A|=(A^*A)^{\frac{1}{2}}$, where $|A|$ is a matrix $B$ such that $B^2=A^*A$. It can verify that $|A^*B|\le ||A|| |B|$ and $|tA|=|t||A|$, for any complex number $t$. Note the relations $|AB|=|A||B|$ and $|A+B|\le |A|+|B|$ are not generally true.
Search for positive elements in C$^*$-algebras.