Q. Given that A is invertible , check whether A + A^(-1) is invertible or not ? A. A is invertible . => A^-1 exists ( sorry about not using the powers directly. i am not able to click on the required icon ). Suppose a non trivial solution exists to ( A + A^(-1) ) X = 0 ( which means lets assume A + A^(-1) is non - invertible ). => a non trivial solution exists to AX = A^(-1) ( -X) or to AX = A A^(-2) ( -X) or to A [ A^(-2) + I ] X = 0 clearly , [ A^(-2) + I ] X = Y is equal to 0 as A is invertible i.e. [ A^(-2) + I ] X =0 We also know that a non trivial solution for X exists => [ A^(-2) + I ] is non - invertible. => A^(-2) must have an eigen value = -1 => A also must have an eigen value = -1 ( by the theorem of eigen value decomp.) => A + A^(-1) is non-invertible only when the matrix A has an eigen value = -1. Hence, generally speaking, it is not invertible. Thanks a lot.