# Properties of Inverse Matrices

## Main Question or Discussion Point

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?

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

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.

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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?

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HallsofIvy