I'm a bit confused how I would solve these problems of matrix inversion. I know the basic properties of inversion, but need some explanation on how to prove or disprove questions such as the ones below: I need to justify each these as true or false. 1. If A is an invertible, real matrix... is the matrix A+2(A^-1) invertible? 2. And also, if a matrix B satisfies: (B^3)-2B+I=0, then the matrix (B-I) cannot be invertible? I would appreciate any help on how to solve these problems, or even if someone could provide me with some properties that I can use to solve them. I know I saw an example asking if A+(A^1) is invertible, given that A is invertible; in this example A+(A^1) was said to be equal to (A^2)+I, and then the problem was solved from here; why are these two equations equal? The extent of what I know about invertible matrices is how to solve for the inverse of A with the formula, and that A(A^(-1))=I=(A^(-1))A, and how to solve inverse matrices using elementary row operations... Thanks for the help!