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Question
I am just wondering what the answer to this question is. I have been trying it for 4 days now and i can't seem to get the right answer.

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

A. (A+B)^2 = A^2+B^2+2AB
B. ABA^-1=B
C. (I_n-A)(I_n+A)=I_n-A^2
D. 8A is invertible
E. (AB)^-1=A^-1B^-1
F. A+A^-1 is invertible
I have gotten B , C , D , F as my answers but my teacher is telling me that they are not the answers. Any help would be greatly appreciated.

A. Not true in general since A and B don't necessarily commute.

B. Not true; need to commute.

C. True: Distributive law works for matrices and IA = AI always.

D. It looks like the number 8 is multiplying A; typo? If it is BA, then (with help from the associative law) (BA)^-1(BA) = A^-1(B^-1B)A =A^-1A = I, so that it is invertible.

E. Not true, however, (AB)^-1 = (B^-1)(A^-1).

F. Not true: try multiplying it by its inverse and seeing if it comes out equal to I.

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#### randy patton

##### Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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