If P= [ array cc 3 / 2 & 1 / 2 -1 / 2 & 3 / 2 array ], A= [ array ll 1 & 1 0 & 1 array ] and Q=P A P^ , then P^ Q²⁰⁰⁵ P is

Step-by-step solution

We have, P^ P= [ array cc 3 / 2 & 1 / 2 -1 / 2 & 3 / 2 array ] [ array cc 3 / 2 & -1 / 2 1 / 2 & 3 / 2 array ]= [ array ll 0 & 0 0 & 1 array ] array lrl & P^ P & =I or P^ =P⁻¹ As, & Q & =P A P^ array P^ Q²⁰⁰⁵ P=P^ [ (P A P^ ) (P A P^ ) 2005 . times ] P= (P^ P ) A (P^ P ) A (P^ P ) (P^ P ) A (P^ P ) _ 2005 times =I A²⁰⁰⁵=A²⁰⁰⁵ Now, A= [ array ll 1 & 1 0 & 1 array ]^ A^2= [ array ll 1 & 1 0 & 1 array ] [ array ll 1 & 1 0 & 1 array ]= [ array ll 1 & 2 0 & 1 array ]A^3= [ array ll 1 & 2 0 & 1 array ] [ array ll 1 & 1 0