Properties of Inverse Matrices

[tigger1989]

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

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?

 

[zhentil]

The ones that are false all have trivial counterexamples. What have you tried so far?

 

[tigger1989]

The ones that are false all have trivial counterexamples. What have you tried so far?

Hi, I'm a little rusty on Linear Algebra, so forgive my ignorance on some of the approaches. Can you elaborate by what you mean by "trivial"?

I tried random matrices (often with easy numbers, such as the identity matrix) for A, B, C, D, E, and F, but only found counterexamples for A, B, and F (but knowing for a fact that C, D, and E are not all correct).

By one of the simple properties of inverse matrices, I am almost certain that C is correct.

 

[trambolin]

C is true, since otherwise A^(-1) is not invertible in the first place, with $A^{-5}=A^{-1}...A^{-1}$ hence if the determinant is zero then one of them has also a zero determinant.

D holds only for reals and from the positive eigenvalues of A^(2) .$trace(A^2)>0, \det(A^2)>0$
$$(A+A^{-1})= (I+A^{2})A^{-1} = (A(I+A^{2})^{-1})^{-1}$$

E follows from the distribution property,

I think I have to sleep so what is the the counterexample for F?

 

[HallsofIvy]

Any two matrices that do not commute will give an counterexample to F.

 

[trambolin]

Haha, Of course! That was lame, sorry!