list manipulation – Removing a random ith row and column from a matrix

Currently I am trying to delete a randomly-chosen $i^{th}$ row and column from a square $n times n$ matrix $A$. So far I come up with the following code:

Drop[A, {RandomInteger[{1, 400}]}, {RandomInteger[{1, 400}]}]

The problem with this command is that the random integer for {i} is not the same as the random integer for {j}.

Is there a way of making them consistent, so that I drop the ith row and corresponding column while maintaining the randomness of selecting $i$?

And if the aim was to not delete the row and column entirely but to replace all their elements with, say 0, how would you go about it?

Thank you.