Programming

I saw that problem a few weeks ago on a youtube thumbnail, it's quite a nice problem and I think it gets even more interesting if you start to consider 5^3 or even 5^N horses. Cool that you actually coded a solution for it! I like the idea of hiding the actual values to prevent the developer from cheating :)

I haven't really checked whether you program does exactly what I'd expect it to do since I had a hard time to understand what's happening. I think a few comments here and there would help to understand what exactly is going on and maybe some better names would hep, too (example: number_matrix - what does it do? I see from the type that it is a 2d array of ints but what does it contain and how does it relate to horse_matrix). I still have some general recommendations for you how you could improve your code:

1. You seem to use OS specific random numbers, there is a C++ api for that nowadays (std::random_device / std::mt19937)
2. You call rand() % 26 and check against 0, in that case you reroll - you could instead directly calculate rand() % 25 + 1
3. As alternative to that manual rand() approach, you could look into std::shuffle
4. Take a look at sets (e.g. std::unordered_set), they have a containts() function that tells you whether a value is in the set or not. It's not a big issue here but using vectors with std::count when all you want to do is check whether an element is in a set will not scale well
5. Consider using an enum instead of the char option. I'd go with something like enum class Option { kRow, kColumn };. If you want to stay with a char, I'd recommend not using a preprocessor directive for this (#define) but instead go with a regular constant (e.g. constexpr char kRow = 'r')
6. Consider using std::unique_ptr<MatrixPosition> instead of returning a raw pointer - that way you avoid memory leaks.
7. ignore_count and ignore_vector are only used in one function, so they don't have to be members of the class. InitializeMatrix probably does not need to be a class in the first place.
8. Consider not using using namespace std. For small projects that's not an issue for bigger ones you'll find your namespace polluted.

Something that you might also not be aware of is that std::endl automatically flushes buffers and makes sure that your line is printed immediately. If you don't need this you can just use \n instead which is a little bit faster. But that's more of a "fun fact" than a general recommendation.

Hope that helps and doesn't sound too harsh, I know how code reviews can feel sometimes..