Determinant of a $3 \times 3$ matrix with element sum $k$

Question

The sum of all the elements of a $3 \times 3$ matrix $A$ is $k$. What is the maximum value of the determinant of $A$?

Answer 1

There should be no maximum value. If we want $\det A = M^3$, then letting: $$A = \begin{pmatrix}M & -3M+k & 0 \\ 0 & M & 0 \\ 0 & 0 & M \end{pmatrix}$$ should have the sum of all elements be $k$, and $\det A = M^3$.