Properties of Inverse Matrices

[tigger1989]

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?

 

[pizzasky]

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

 

[Architeuthis Dux]

C is actually correct. A^5 is invertible because if A is invertible, then A^-1 exists and (A^-1)^5 is also invertible. This means that A^5 must also be invertible.

D is incorrect. A+A^-1 is only guaranteed to be invertible if A^-1 is its own inverse, which is not always the case.

E is also incorrect. While the formula looks similar to the formula for finding the inverse of a matrix, it is not always true. For example, if A=I, then (I+n)(I+I^-1)=2I+I+I^-1=4I, which is not necessarily the inverse of A. Therefore, this formula does not hold for all invertible matrices A and B.