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 & C= A ^ 2^ ( n -2) then which of the following statements are true?

Step-by-step solution

aligned & B=A^ 2^n =A^ 2 2^ n-1 = (A^2 )^ 2^ n-1 = (A⁻¹ )^ 2^ n-1 = (A^ 2^ n-1 )⁻¹= (A^ 2 2^ n-2 )⁻¹= ( (A^2 )^ 2^ n-2 )⁻¹ & = ( (A⁻¹ )⁻¹ )^ n^ n-2 =A^ 2^ (n-2) aligned So, B = C aligned & (B-C)= null & (B+C)(B-C)=0 & det .(B-C)=0 aligned