Do These Matrix Inversion Properties Always Hold?

[thepassenger48]

Homework Statement

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

1. (A+A^-1)^7 = A^7+A^-7
2. 2A is invertible
3. (A+B)² = A² + B² + 2AB
4. (ABA^-1)^6 = AB^6A^-1
5. A + B is invertible
6. ABA^-1 = B


I know 2A should be invertible, and number 3 wrong, but what else?
Thanks

 

[radou]

Try to think of examples if a fact bothers you or if you can't proove it.

 

[HallsofIvy]

For 1) and 3, write out the computation- for 1 you might want to look at (A+ A^-1)² first.

For 5, if A is invertible, then so is B= -A.

 

[D H]

Think of whether 1 and 5 are true for all invertible 1x1 matrices (i.e., the non-zero reals or non-zero complex numbers).

For 4, expand the left-hand side.

For 6, what are the conditions that a matrix B be self-similar with respect to A?