# Properties of Inverse Matrices

## Homework Statement

Determine which of the formulas hold for all invertible nxn matrices A and B

A. AB=BA
B. (A+A^–1)^8=A^8+A^–8
C. A^5 is invertible
D. A+A^–1 is invertible
E. (In+A)(In+A^–1)=2In+A+A^–1 (where In is the identity matrix)
F. (A+B)^2=A^2+B^2+2AB

## Homework Equations

Certain properties of inverse matrices can be used. For example, if A is invertible, then A^k is invertible for all k greater or equal to 1 (this proves C to be correct).

## The Attempt at a Solution

I was able to find counterexamples to prove A and B and F incorrect. However, the webwork program (designed for practicing basic Linear Algebra) I am using states that C, D, and E are not all correct ... what am I missing?

Option D is not always true. Try to find a matrix A where $$A^{-1} = -A$$.