Question

Let $A,B,C$ be three square matrices of same order satisfying the equations $A^2=A^{–1},$ $B=A^{2^n}$ and $C=A^{2^{(n-2)}}.$ Then which of the following statements is(are) TRUE ?
(Here, $O$ is a null matrix of the same order.)

• A. $det(B-C)=0$
• B. $(B+C)(B-C)=O$
• C. $B$ must be equal to $C$
• D. None of these

Answer 1

Answer: (A), (B), (C)

$B$ must be equal to $C$

$B=A^{2^n}=A^{2\cdot 2^{n-1}}=(A^2{)}^{2^{n-1}}$
$=(A^{-1}{)}^{2^{n-1}}=(A^{2^{n-1}}{)}^{-1}=(A^{2\cdot 2^{n-2}}{)}^{-1}$
$=\left(\right(A^2{)}^{2^{n-2}}{)}^{-1}=\left(\right(A^{-1}{)}^{-1}{)}^{2^{n-2}}=A^{2^{n-2}}=C$
So, $B=C$
$\Rightarrow B-C=O$
$\therefore det(B-C)=0$

We have, $B-C=O$
$\therefore (B+C)(B-C)=O$