phonetically | Spell words phonetically with ease | Data Manipulation library

I have a matrix with similarity scores that looks like this:

I need to find the highest possible sum of scores in this matrix. The sums have to fulfill a condition though:

If a number has been used for the sum, no numbers of its row or column or any previous rows or columns can be used for the sum anymore. This is because the order of the names matters.

I can start at any number, but all the values to the left and above that value will then be disqualified for the rest of that sum, as well as the values in the same row and column.

The highest possible sum with this system is 130 (10 + 100 + 10 + 10). That's the number i want in the end.

My strategy at the moment is to calculate all possible sums, and then simply select the highest. But how do i code the condition that i described above? Does it make sense?

Here are more examples of allowed (green) and not allowed (red) sums:

Another example of a valid sum:

1. I start at the upper left corner. I choose 10. I can not add the 12 or the 11 that are in the same column anymore.
2. I choose one of the remaining numbers. 12. I can no longer choose 100, 11, 10, or 25 from that column and row (and the previous columns and rows). I can now only choose 22 or 10 for the last number to add.
3. If i pick 22, my total sum is 44. If i pick 10, my total sum 32.

The reason i'm using this system is because i'm trying to create an algorithm that compares full names of people and assigns it a probability that they're the same person - purely based on name information.

My current code looks like this: