"How else would you solve Ax = b for x, if not by computing the inverse of A (assuming A is a 3x3 or 4x4)?" For instance, there a fast iterative solvers. They

"How else would you solve Ax = b for x, if not by computing the inverse of A (assuming A is a 3x3 or 4x4)?" For instance, there a fast iterative solvers. They start with an initial guess for the solution (which can be an arbitrary vector) and successively generate more accurate approximations. These iterative solvers don't need access to the inverse of A. Furthermore, many of those solvers don't even need access to A: They only need access to a subroutine which, given a vector u, returns Au. Many matrices arising in applications (for instance one often reduces the problem of solving partial differential equations to the problem of solving a (very huge) linear system of equations) are sparse (have only few non-zero entries). For these it can make sense to write specialized code which specifically calculates Au. If you want to read more about this, then a good starting point is https://en.wikipedia.org/wiki/Kaczmarz_method https://en.wikipedia.org/wiki/Kaczmarz_method.