I have a problem about matrices, but the problem is that I don't know enough about them to know the answer (and I am not a native speaker, so I'll try my best to describe the problem).
So, imagine that I have various metal detectors (A, B, C) that can detect (or not) several metals. For example:
| Metal | Detector A | Detector B | Detector C |
|---|---|---|---|
| Iron | 1 | 1 | 1 |
| Cobalt | 1 | 0 | 1 |
| Nickel | 0 | 0 | 1 |
1 means that the detector is going to emit a sound when you aproach that metal, and 0 means that the detector will not emit any sound to that metal. So, if someone ask me, are these detectors enough to distinguish between those metals? the answer is yes, because if you imagine the previous table as a matrix, then its determinant is not equal to zero (they make an orthonormal/orthogonal basis I think? It's about 10 years the last time I took a maths class).
The thing is that I have a lot of detectors and a lot of metals, so if I want to know which detectors are enough to distinguish between (for example) 5 metals, I have to choose 5 detectors -> calculate the determinant, choose another 5 detectors -> determinant, and again and again until I find a suitable set of detectors. You get the idea. Doing an excel about this is a pain in the ass.
So my problem is, imagine that I have a matrix (let's call it M) with all the detectors and metals that I have (it is not neccesarily a square matrix). Is there any mathematical operation that, applied to M, it produces another matrix that will tell you, for example, with those detectors that you consider in the matrix M you can distinguish between iron, cobalt and nickel, but you cannot distinguish between iron and copper, or copper and titanium?
Or a mathematical operation applied to that matrix M that tells you, you need these detectors A B and C if you want to distinguish between Iron, Cobalt and Nickel? Or if you consider copper and those previous 3 metals you will need the detector X?
I hope that my problem is well stated. I don't have a high level of maths, but if you point me into the right direction I will look into that. Thank you.