Let A= [ array cc 1 2 & -2 0 & 1 array ] and P= [ array cc & - & array ], 0 . If B = PAP ^ T , C = P ^ T B ¹⁰ P and the sum of the diagonal elements of C is m n , where gcd ( m , n )=1 , then m + n is :

Step-by-step solution

aligned & P = [ array cc & - & array ] & P ^ T P = I & ~B = PAPT aligned Pre multiply by P ^ T ( Given) P ^ T ~B = P ^ T P AP ^ T = AP ^ T Now post multiply by P P ^ T BP = AP ^ T P = A A ^2= P ^ T ~B ^2 P Similarly A¹⁰=P^T B¹⁰ P=C aligned & A= [ array cc 1 2 & -2 0 & 1 array ] (Given) & A^2= [ array cc 1 2 & - 2 -2 0 & 1 array ] aligned Similarly check A^3 and so on since C=A¹⁰ Sum of diagonal elements of C is ( 1 2 )¹⁰+1 aligned & = 1 32 +1= 33 32 = m n & g~cd ( m , n )=1( Given ) & m + n =65 aligned