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Select all statements below which are true for all invertible n×n matrices A and B?

hi iv been stuck on this questions for a long time. if anyone can help id be delighted thanks

Select all statements below which are true for all invertible n×n matrices A and B

A. A2 is invertible

B. (ABA−1)7=AB7A−1

C. (A+B)2=A2+B2+2AB

D. ABA−1=B

E. (AB)−1=A−1B−1

F. A+B is invertible

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The Best Answer for Select all statements below which are true for all invertible n×n matrices A and B?

A) True, since |A| ≠ 0 implies that |A^2| = |A|^2 ≠ 0.

B) True (for any exponent instead of 7 as well).

C) False, since AB ≠ BA in general:

(A + B)^2 = (A + B)A + (A + B)B = A^2 + BA + AB + B^2.

D) False, since this is equivalent to AB = BA which is not true in general.

E) False; the actual formula is (AB)^(-1) = B^(-1) A^(-1), since

(AB)(B^(-1) A^(-1)) = A (BB^(-1)) A^(-1) = AIA^(-1) = AA^(-1) = I.

F) False; A = I and B = -I are invertible, but A + B = 0 is clearly not invertible.

I hope this helps!

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