Paragraph: Answer the following by appropriately matching the lists based on the information given in the paragraph. Let f(x) is defined as f(x)= | array ccc (x+ ) & (x+ ) & (x+ ) (x+ ) & (x+ ) & (x+ ) ( - ) & ( - ) & ( - ) array | and f(9)= 0 and P= [ array cc ( / 9) & ( / 9) - ( / 9) & ( / 9) array ] , where , and be non-zero numbers such that ( P^6+ P^3+ I ) is the zero matrix and where I is an identity matrix of

Step-by-step solution

aligned & f^ (x)=0 f(x)= constant f(9)= & _ k=1 ^9 f(k) f(9) = 9 =9 & P^2=P P= [ array cc (2 / 9) & (2 / 9) - (2 / 9) & (2 / 9) array ] & P^n= [ array cc (n / 9) & (n / 9) - (n / 9) & (n / 9) array ] & P^6+ P^3+ I= [ array cc -1 / 2 & 3 / 2 - 3 / 2 & -1 / 2 array ]+ [ array cc 1 / 2 & 3 / 2 - 3 / 2 & 1 / 2 array ]+ [ array ll 1 & 0 0 & 1 array ]=0 & aligned aligned & - 2 + 2 + =0 and 3 2 ( + )=0 =- & =a & array llll & (II) P ; & (III) P and & (IV) Q array & aligned