Chapter 8.1, Problem 47E

Let us first introduce two notations.
For a complex $n\times n$ matrix A, let $\left|A\right|$ be the matrix whose ijth entry is $\left|a_{ij}\right|$ .
For two real $n\times n$ matrices A and B, we write $A\le B$ if $a_{ij}\le b_{ij}$ for all entries. Show that
a. $\left|AB\right|\le \left|A\right|\left|B\right|$ , for all complex $n\times n$ matrices A and B, and
b. $\left|A^t\right|\le \left|A^t\right|$ , for all complex $n\times n$ matrices A and all positive integers t.