graphs – Changing a matrix to become an ancestry matrix

An ancestry matrix $M$ for rooted tree $T$ is defined as $M(ij) = 1$ iff node $i$ is an ancestor of node $j$. Suppose we are given a matrix $X$. We can easily check that if $X$ is compatible with some rooted tree(and create the tree) or not.
The question is if $X$ is not an ancestry matrix, how hard is the problem of modifying elements of $X$ so that it becomes an ancestry matrix.

Note: Stated problem can be seen as using hamming distance on the matrices. A related problem would be for what metrics on the matrices this problem can be solved eficciently?