&quot;How else would you solve Ax = b for x, if not by computing the inverse of A (assuming A is a 3x3 or 4x4)?&quot;<p>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.<p>These iterative solvers don&#x27;t need access to the inverse of A. Furthermore, many of those solvers don&#x27;t even need access to A: They only need access to a subroutine which, given a vector u, returns Au.<p>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.<p>If you want to read more about this, then a good starting point is <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Kaczmarz_method" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Kaczmarz_method</a>.