Problem
Consider a generic matrix $A$, we are going to think of a simple case by taking into consideration a $3 \times 3$ matrix:
$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} & a_{2,2} & a_{2,3}\\ a_{3,1} & a_{3,2} & a_{3,3}\\ \end{pmatrix} $
Consider now having $A'$ as:
$ A' = \begin{pmatrix} a'_{1,1} & a'_{1,2} & a'_{1,3}\\ a'_{2,1} & a'_{2,2} & a'_{2,3}\\ a'_{3,1} & a'_{3,2} & a'_{3,3}\\ \end{pmatrix} $
The following holds:
$a'_{i,j} \leq a_{i,j}$
Question
I would like to know if the following:
$|A'| \leq |A|$
If it holds, can you prove it?
Another problem
What if we considered:
$ a_{i,j} \leq 1, a'_{i,j} \leq 1 $
Considering also that $A$ is a stochastic matrix?
This does not mean that both $A$ and $A'$ are stochastic. I am considering $A$ stochastic and $A'$ obtained as a reduced version of $A$ so that $A'$ is not stochastic but its values are all between 0 and 1.