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Determine which of the formulas hold for all invertible nxn matrices A and BA. (A+B)^2...

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bobby9901 | Student, College Freshman | (Level 1) Salutatorian

Posted April 8, 2013 at 6:41 AM via web

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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

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted April 11, 2013 at 12:16 AM (Answer #1)

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`(A+B)^2=A^2+AB+BA+B^2,` which is usually not equal to `A^2+2AB+B^2.`so choice A is not true for arbitrary invertible matrices.

B. If `ABA^(-1)=B,` then `AB=BA.` This is usually not true, so choice B is not true for arbitrary invertible matrices.

C. `(I_n-A)(I_n+A)=I_n^2+I_nA-AI_n-A^2=I_n-A^2,` so choice C is true.

D. If `A` is invertible, then `det A!=0,` and `det 8A=8detA!=0,` so `8A` is invertible. Choice D is true.

` ` E. `(AB)^(-1)=B^(-1)A^(-1).` In general, this will not equal `A^(-1)B^(-1),` so choice E is not true for arbitrary invertible matrices.

F. It is easily checked that `A=[[0,1],[-1,0]]` is invertible, and `A+A^(-1)=0,` which is not invertible. Choice F is not true for arbitrary invertible matrices.

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pramodpandey | College Teacher | (Level 3) Valedictorian

Posted April 10, 2013 at 7:29 AM (Answer #1)

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B,C, D, and F

explanation for F

Let `A=[[1,2],[0,1]]`

`A^(-1)=[[1,-2],[0,1]]`

`A+A^(-1)=[[2,0],[0,2]]=C (say)`

`C^(-1)=[[2,0],[0,2]]`

`` similarly otherpart can be explain.

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