At the edge of madness, we finally see that our comrade Ax=b can help us succeed through Gauss-Jordan elimination. Row operations allow us to travel out of the darkness, from $[\, \mathbf{A}\, |\, \mathbf{I}\, ]$ to $[\, \mathbf{I}\, |\, \mathbf{A^{-1}}\, ]$, with a full 2x2 example and the plan for 3x3s and up.