I had linear algebra a long time ago, and I feel I can do all the computations. However, I still don't know why, for instance, a determinant tells me a system of equations is solvable. I don't know why the determinant gives me the volume of a parallelepipeds.

Most textbooks on linear algebra seem to just skim over this part. The books may tell you why a result is true for the 2x2 matrix, but not the general case.

What book should I read to learn the logic behind concepts like the determinant?