Here $I3$ is the best example for such matrix.
$A = \begin{bmatrix} 1 & 0 & 0\\ 0& 1 & 0\\ 0& 0 & 1 \end{bmatrix}$
Here even number entries are 6 and $|A|=1$.We can't get more than 6 even element for this type of matrix.
Hence,Maximum number of even entries in A is $6$.