It's a little surprising, in a "no-bullshit" discussion of "theoretical and computational aspects of linear algebra," to see matrix inversion touted as the way

It's a little surprising, in a "no-bullshit" discussion of "theoretical and computational aspects of linear algebra," to see matrix inversion touted as the way to solve linear equations. The guide literally introduces the examples by saying "Dude, enough with the theory talk, let's see some calculations." Yet standard numerical practice avoids literal inversion, in favor of factorization methods. E.g., It is common practice to write the form A^{-1}b for economy of notation in mathematical formulas... The trouble is that a reader unfamiliar with numerical computation might assume that we actually compute A^{-1}... On most computers it is always more effective to calculate A^{-1}b by solving the linear system Ax = b using matrix factorization methods... (Dennis & Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations", section 3.2). E.g., As a final example we show how to avoid the pitfall of explicit inverse computation... The point of this example is to stress that when a matrix inverse is encountered in a formula, we must think in terms of solving equations rather than in terms of explicit inverse formation. (Golub and Van Loan, "Matrix Computations", section 3.4.11).