Properties of Inverse Matrices

AI Thread Summary
The discussion focuses on the properties of inverse matrices, specifically evaluating which formulas hold for all invertible nxn matrices A and B. It is established that A^5 is always invertible, confirming option C as correct. Counterexamples are provided to demonstrate that options A, B, and F are incorrect. Additionally, it is noted that option D is not universally true, particularly when considering matrices where A^(-1) equals -A. The conversation emphasizes the need for careful consideration of specific matrix properties to determine the validity of these formulas.
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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?
 
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Option D is not always true. Try to find a matrix A where A^{-1} = -A.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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