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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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High School Teacher
`(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.
Posted by degeneratecircle on April 11, 2013 at 12:16 AM (Answer #1)
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