Homepage › Solution manuals › Gilbert Strang › Linear Algebra and Learning from Data › Exercise 3.1.6
When U = V = In, equation (9) gives formula (A − I)−1 = A−1 + A−1(I − A−1)−1A−1
We have $(A-I)^{-1} = (A − AA{-1} ) {−1} = (A(I − A{-1}) ) {−1} = (I−A{-1}){−1}A^{−1} = (I−A^{−1}+A^{−1})(I−A{-1}){−1}A^{−1} = A^{−1}+A^{−1}(I−A{-1}){−1}A^{−1}$
