This question came in my practice paper
If there are three square matrix A, B, C of same order satisfying the equation $A^2=A^{-1}$ and let $B=A^{2^n}$ and $C=A^{2^{(n-2)}}$ then which of the following statements are true?
(A) det. (B – C) = 0
(B) (B + C)(B – C) = 0
(C) B must be equal to C
(D) none
I did found out the answer by substituating the value of n =2 and 3 and then soving the equations But I wanted a proof of it by solving in terms of n (i.e without value substituation)
Every Help is welcomed
Regards!
$$A^2=A^{-1}$$ $$A^3=I$$ $$B=A^{2^n}=A^4A^{2^{n-2}}=AC$$ A:
$$B-C=(A-I)A^{2^{n-2}}$$ Now, powers of A are invetible and not $0$, and $A-I$ can be non-zero. B:
$$(B+C)(B-C)=(A^2-I)A^{2^{n-1}}$$ Now, powers of A are invetible and not $0$, and $A^2-I$ can be non-zero.