Invertible Matrices Question, please help!?
Select all statements below which are true for all invertible nxn matrices A and B:
A) (AB^-1) = (A^-1)(B^-1)
B) (In - A)(In + A) = In - A^2 ***
C) A + B is invertible
D) (A+B)^2 = A^2 + B^2 + 2AB
E) (A^8)(B^6) is invertible
F) AB = BA
***In is the identity matrix
I think I know that A and C are true but I don't know about the others...any ideas as to what the correct ones are would be greatly appreciated!
Thanks a million!
A) is False. (AB)^(-1) = B^(-1)A^(-1) NOT A^(-1)B^(-1)
B) is true. The product of A and I commute.
C) is false. Take as a simple counter example A = I and B = - I. Then A and B are both invertible, but there sum most certainly isn't.
D) False! (A + B)2 = A2 + B2 + AB + BA but these latter terms can not be combined in general because matrix multiplication does not commute.
E) This is true. det(A^8B^6) = (det(A))^8 (det(B))^6 which is nonzero because det(A) and det(B) are nonzero.
F) False, matrix multiplication doesn't commute.
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A) Meant to be (AB)^-1 = (A^-1)(B^-1). Test with 2 different invertible matrices NOT TRUE
B) Matrix multiplication (A+B)(C+D)= AC+AD+BC+BD so this is OK
C) Consider A=In (2X2) and B = -In both of which are invertible, but A+B s the zero matrix and
is not invertible. So not true
D) See B) matrix multiplication is not commutative i.e AB does not (in general) =BA. So NO
E) DET[(A^8)(B^6)} = DET(A^8)DET(B^6) = [DETA]^8[DETB]^6
F) Check with 2 different invertible matrices, you will find it not true