The guide
A matrix is a function. It takes a vector and returns another one, and because the function is linear it is completely described by what it does to the basis vectors. That is the whole idea, and the arithmetic everyone remembers from school is just the bookkeeping for it.
Once you see it that way the rest follows. A matrix product is composition, two functions applied in turn. The determinant is how much the function scales area or volume, so a determinant of zero means the function squashed a dimension flat and cannot be undone. Eigenvectors are the directions the function leaves pointing where they were, only stretched.
You are implementing these by hand because every model later is a pile
of them, and reading X @ W should feel like reading a sentence.