> In which bit of math do people actually instantiate an inverse? I am assuming you are being rhetorical here. I also think your working hypothesis is that the

> In which bit of math do people actually instantiate an inverse? I am assuming you are being rhetorical here. I also think your working hypothesis is that the only use of a matrix inverse is to solve a linear system of equations. Under this hypothesis it does make sense that there is no utility to computing inverses. However, this hypothesis is incorrect. There are tons of cases where you would want to get at the entries of an inverse, but has nothing to do with solving equations. Here area few examples: consider data generated from a multivariate Gaussians. Now you want to know which components are independent of each other. This is given by whether the inverse of the correlation matrix is zero or not. Another example, say you want to minimize a function defined over matrices, but want to minimize it only over the set of positive semi-definite matrices. One way of doing these is to use a barrier function: a function that blows up astronomically when you approach the boundary of your constraint set. For positive definite matrices such a barrier function is the logarithm of the determinant. I could go on...